VERSITY  OF  CALIFORNIA,  SAN  DIEG 


1822023445505 


3  1822  02344  5505 


INTRODUCTION 


NATURAL   PHILOSOPHY 


DESIGNED  AS 


A   TEXT   BOOK, 

FOR 

THE   USE   OF  THE   STUDENTS 

I 

ra 

YALE  COLLEGE. 

COMPILED    FROM    VARIOUS    AUTHORITIE 


BY  DENISON  OLMSTED,  LL.D., 

PROFESSOR  OF   NATURAL  PHILOSOPHY   AND   ASTRONOMY. 


STEREOTYPE    EDITION. 


NEW  YORK: 
PUBLISHED  BY    ROBERT   B.    COLLINS, 

254  PEARL-STREET. 

1853. 


Entered,  according  to  Act  of  Congress,  in  the  year  1344,  ttf 

DENISON  OLMSTED, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut 

* 


Bwreotyped  by 

RICHARD  C.  VALENTIN?, 
41  Gold  Street,  Niw  York. 


ADVERTISEMENT  TO  THE  FIFTH  EDITION 


THE  publishers  of  this  work  have  hitherto  forborne  to  stereotype  it,  in 
order  that  the  author  might  more  conveniently  make  such  alterations  and 
corrections,  in  the  successive  editions,  as  his  own  studies,  and  the  commu- 
nications of  his  scientific  friends  who  use  the  work  as  a  text-book,  might 
suggest.  Of  these  means  of  improvement,  he  has  diligently  and  care- 
fully  availed  himself;  and  having  now  brought  the  work  to  the  highest 
degree  of  improvement  and  accuracy  that  seems  at  present  attainable  by 
him,  he  has  consented  that  it  should  be  stereotyped,  without  contempla- 
ting any  further  changes  for  some  time  to  come. 

The  more  economical  mode  of  publishing  thus  secured ;  the  adoption 
of  a  style  of  printing  somewhat  more  compact,  while  it  is  equally  legible, 
thus  rendering  it  convenient  to  reduce  the  two  volumes  to  one ;  and,  above 
all,  the  extensive  and  increasing  demand,  have  enabled  the  publishers 
materially  to  reduce  the  price  of  the  work,  in  conformity  with  what  is 
believed  to  be  the  general  wish  of  the  numerous  literary  institutions  that 
use  it  as  a  text-book. 

Although  some  passages  found  in  former  editions  have  been  omitted 
in  the  present,  yet  others  deemed  of  greater  value  have  been  added,  so 
that  the  amount  of  matter  in  the  present  form  has  not  been  diminished, 
but  rather  increased. 

From  many  of  his  brethren  of  other  Colleges,  the  author  has  received 
corrections  and  suggestions,  for  which  he  is  indebted  in  no  small  degree 
for  the  more  improved  and  accurate  state  of  the  work  as  now  published. 
To  all  of  them  he  would  express  his  grateful  acknowledgments.  To 
Professor  LOOMIS  and  Professor  SNELL,  he  is  under  peculiar  obligations. 

To  blend  practical  utility  with  scientific  accuracy,  was  originally,  and 
still  continues  to  be,  the  aim  of  this  work — a  plan  which  secures  to  the 
scholar,  along  with  the  discipline  of  the  understanding,  habits  of  philo- 
sophical reasoning  and  observation  on  the  phenomena  of  nature  and  art, 
and  information  which  may  be  made  available  in  the  practical  business 
of  life. 

YALE  COLLEGE,  May,  1844. 


PREFACE. 


THE  compiler  of  this  work  has  had  two  objects  constantly  in  view  first,  to  make 
the  student  thoroughly  and  familiarly  acquainted  with  the  leading  principles  of  Natural 
Philosophy ;  and  secondly,  to  furnish  him  with  as  much  useful  information  as  possible, 
within  so  limited  a  compass.  The  foundation  of  all  accurate  attainments  in  Natural 
Philosophy  and  Astronomy  being  laid  in  the  science  of  Mechanics,  a  large  proportion 
of  the  work  is  devoted  to  this  subject.  The  Mathematical  Elements  of  Mechanics  are 
moreover  first  considered,  separately,  that  the  student  may  have  nothing  to  divert  his 
mind  from  the  contemplation  of  the^g  universal  and  fundamental  truths.  Still  further 
to  render  the  knowledge  of  these  truths  familiar,  as  well  as  to  supply  a  most  useful  intel- 
lectual exercise  to  the  student,  a  great  variety  of  Problems  are  annexed  to  each  chapter, 
the  utility  of  which  must  be  obvious  to  every  experienced  instructor.  Indeed,  Problems 
hold  so  important  a  place  in  the  estimation  of  the  writer,  that  he  has  introduced  them  into 
various  parts  of  the  work,  wherever  the  subject  appeared  to  be  susceptible  of  deriving 
aid  from  them.  Problems  put  the  student  on  his  own  resources ;  they  compel  him  to 
think  for  himself;  they  lead  him  to  a  just  understanding  of  the  principles  demonstrated ; 
and  they  teach  him  how  to  reduce  his  knowledge  to  practice.  These  truths  are  so 
obvious,  that  it  is  difficult  to  account  for  the  singular  fact,  that  treatises  on  Natural 
Philosophy  have,  in  general,  contained  few  or  no  problems,  although  they  occupy  so 
large  a  space  in  most  of  the  branches  of  the  pure  mathematics. 

The  first  part  of  the  following  treatise  on  mechanics,  comprising  the  "  Mathematical 
Elements,"  is  taken  chiefly  from  Bridge's  Mechanics.  This  work  was  peculiarly 
adapted  to  our  purpose,  partly  because  it  is  written  in  a  style  well  suited  to  the  average 
capacities  and  attainments  of  college  classes,  and  partly  because  it  is  enriched  with  a 
finer  collection  of  problems  than  any  similar  work  with  which  we  are  acquainted. 
We  have  aimed  to  select  such  parts  as  promised  the  most  practical  utility  ;  and  in 
order  to  adapt  the  treatise  to  the  purposes  of  recitation,  the  propositions  are  more 
distinctly  enunciated  than  in  the  original  work,  and  various  alterations,  and  occasional 
additions,  are  introduced,  and  explanations  added  by  note  or  otherwise,  with  the  view 
of  suiting  it  better  to  the  course  of  instruction  adopted  in  Yale  College. 

In  Part  II,  the  Practical  Applications  of  the  Principles  of  Mechanics  to  the  Arts 
and  to  the  Phenomena  of  Nature,  are  pursued  as  far  as  our  limits  would  permit,  and 
further  perhaps  than  some  instructors  will  deem  necessary ;  for  we  are  aware  that 
some  maintain  the  expediency  of  occupying  the  attention  of  the  student  almost  exclu- 
sively with  general  principles,  and  leaving  him  to  make  the  application  for  himself. 
According  to  our  experience,  however,  the  student  who  is  furnished  with  the  knowledge 
of  abstract  principles  merely,  seldom  acquires  the  necessary  readiness  in  reducing  them 
to  practice.  It  has  appeared  to  us  no  less  necessary  to  initiate  the  learner  in  the  habit  of 
philosophizing,  than  in  the  doctrines  of  philosophy.  In  this  manner,  he  will  indeed 
acquire  the  knowledge  of  fewer  principles  ;  but  he  will  know  much  better  how  to  use 
his  acquisitions. 

We  cannot,  however,  agree  with  those  instructors  who  have  yielded  to  the  spirit  of 


VI  PREFACE. 

the  age,  (which  is  still  hunting  after  a  "  royil  road"  to  knowledge,)  so  far  as  to  forsake 
demonstration  altogether,  and  substitute  for  the  mathematical  elements  of  Natural 
Philosophy,  text-hooks  in  which  the  principles  of  the  science  rest  on  no  better  basis 
than  mere  popular  illustration,  although  works  of  this  kind  may,  indeed,  furnish  us 
'  with  very  useful  materials  for  exhibiting  the  applications  of  these  principles.  In 
philosophy,  as  in  morals,  the  most  important  principles  are  usually  characterized  by 
great  simplicity :  in  this  respect  the  "golden  rule"  and  the  law  of  gravitation  resemble 
each  other. 

The  subsequent  parts  of  the  work  are  compiled  from  various  authors.  In  the  belief 
that  the  truths  most  important  to  be  inserted  in  a  text -book  on  Natural  Philosophy, 
are,  in  general,  such  as  have  long  been  known,  no  effort  has  been  made  to  conform  the 
style,  either  of  the  propositions  or  the  demonstrations,  to  a  modern  dress ;  but  authors, 
old  and  new,  English  and  French,  have  been  consulted  and  used  indiscriminately. 
Aiming  solely  at  preparing  such  an  elementary  work,  as  would  be  most  useful  to  the 
academic  student,  we  have  not  aspired  to  the  praise  of  originality,  nor  felt  at  liberty 
to  consult  the  pride  of  authorship.  Among  the  truths,  however,  found  in  the  wide 
field  of  Natural  Philosophy,  a  vast  difference  exists  in  regard  to  their  value  ;  and  no 
small  acquaintance  and  familiarity  with  the  subj^t  is  requisite  in  the  writer  of  a  text- 
book, in  order  that  he  may  be  able  to  cull  the  choicest  truths,  and  present  them* to  the 
young  learner  in  their  native  beauty  and  simplicity.  For  this  purpose,  no  powers  of 
original  investigation,  or  gifts  of  genius,  can  compensate  for  the  want  of  the  expe- 
rience of  the  instructor. 

Since  the  publication  of  the  first  edition  of  this  work,  the  compiler  has  been  favored 
with  the  opinions  of  a  number  of  his  brethren  of  other  Colleges,  who  have  used  it  as 
a  text-book.  A  few  would  prefer  to  have  a  more  strictly  mathematical  complexion 
preserved  throughout,  in  the  place  of  those  parts  which  are  written  in  a  more  popular 
style,  since  the  method  of  expressing  philosophical  truths  by  mathematical  formulae, 
is  more  concise  and  comprehensive,  and  better  suited  to  mental  discipline,  and  the 
cultivation  of  a  mathematical  taste.  A  greater  number,  however,  of  those  who  hare 
been  so  good  as  to  communicate  their  views  to  the  writer,  have  approved  of  the  method 
adopted, — namely,  to  establish  the  general  principles  of  the  science  by  rigorous 
mathematical  demonstration,  or  by  precise  experiments,  but  to  rehearse  the  applications 
of  those  principles  to  the  arts,  and  to  the  phenomena  of  nature,  in  a  style  divested,  as 
far  as  possible,  of  technical  phraseology.  They  believe  that  such  a  method  renders 
the  study  of  Natural  Philosophy  peculiarly  attractive  to  the  young  learner,  and  con- 
ducive to  the  formation  of  habits  of  philosophical  observation,  while  they  rely  more 
on  other  parts  of  the  scientific  course,  particularly  on  the  pure  mathematics,  to  fulfil 
the  purposes  of  mental  discipline,  and  to  inspire  a  mathematical  taste.  We  have 
desired  to  accomplish,  as  far  as  possible,  these  several  objects  of  a  philosophical  edu- 
cation,— to  improve  the  faculties  of  the  mind  itself,  to  imbue  it  with  a  love  of  rigorous 
demonstration,  and  to  commence  the  formation  of  habits  of  philosophical  observation, 
which  shall  be  carried  forward,  beyond  the  pale  of  academic  study,  to  be  confirmed 
and  strengthened  throughout  the  period  of  after  life.  The  variety  of  subjects  compre- 
hended under  Natural  Philosophy,  some  admitting  of  strict  geometrical  and  analytical 
reasoning,  and  others  conducted  wholly  by  experimental  research,  is  well  adapted  to 
the  attainment  of  these  important  objects  ;  and  it  is  the  prerogative  of  this  science, 
at  once  to  enlarge  the  mind  by  the  most  profound  inquiries,  and  to  conduct  it  through 
the  most  delightful  and  varied  fields  of  experiment  and  observation. 

In  a  work  necessarily  so  limited  as  this,  (when  compared  with  the  vast  extent  of 
the  subject,)  many  topics  must  be  treated  with  extreme  conciseness,  and  many  others, 
essential  to  a  complete  philosophical  education,  must  be  omitted  altogether.  While 


PREFACE.  Vli 

we  aim  to  furnish  the  student  with  a  knowledge  of  the  great  laws  of  nature,  and  to 
exemplify  and  illustrate  them  by  numerous  applications,  we  can  claim  nothing  more 
than  en  "  Introduction  to  Natural  Philosophy."  suited  to  beginners. 

It  is  recommended  to  the  student  to  make  a  free  use  of  the  Analysis,  especially  in 
reviewing.  Let  him  submit  each  of  the  particulars  indicated  in  this  outline,  to 
deliberate  and  repeated  reflection,  and  he  will  not  only  fully  possess  himself  of  the 
contents  of  the  work,  but  will  lay  up  hi  the  mind  a  system  of  heads,  or  "  common- 
places," under  which  he  can  conveniently  and  usefully  arrange  all  his  future  acquisi- 
tions on  similar  subjects. 

The  course  of  instruction  in  Natural  Philosophy  pursued  in  Yale  College,  to  which 
this  treatise  is  adapted,  proceeds  as  follows.  The  mathematical  part  of  mechanics  is 
first  recited,  in  the  same  manner  as  a  branch  of  the  pure  mathematics.  With  the 
practical  part  commences  a  series  of  familiar  Lectures,*  designed  to  amplify  the  text, 
and  to  illustrate  it  by  numerous  experiments.  These  are  continued  during  the  perusal 
of  the  remainder  of  the  work.  To  the  same  class  is  afterward  delivered  a  select 
course  of  Lectures,  which  are  chiefly  devoted  to  the  discussion  of  the  great  principles 
of  Philosophy  and  Astronomy,  and  especially  to  such  subjects  as  require  a  fuller  atten- 
tion than  they  can  receive  in  the  elementary  course. 


*  Should  any  teacher  who  may  use  this  work,  think  it  better  to  connect  the  Lectures  and  Prtctical 
Applications  immediately  with  the  theoretical  part,  it  will  be  easy  to  do  so,  by  giving  occasional  leaf  JOB  ra 
Part  II,  while  the  student  is  reading  Part  I. 


ANALYSIS. 


IGP  These  Outlines  are  intended  to  aid  in  reviewing,  (answering  the  same  pur- 
pose as  a  seri&s  of  questions,)  and  it  is  earnestly  recommended  to  the  Learner 
to  make  free  use  of  them 


PRELIMINARY    DEFINITIONS. 

Natural  Philosophy  defined,     .         .17 

The  term  Law 17 

Natural  Philosophy  distinguished  from 

Chemistry, 17 

Branches  of  Natural  Philosophy,  .  17 
Mechanics  defined,  .  .  .  .17 
Body  and  Force,  .  .  .  .17 
Statics  and  Dynamics,  .  .  .17 
Hydrostatics  and  Pneumatics,  .  .18 
Two  essential  properties  of  matter,  .  18 
The  term  particle,  .  .18 
Mass,  volume,  density,  defined,  .  18 
Other  properties  of  matter,  .  .  18 
Mechanics  distinguished  from  Geom- 
etry,   18 

MOTION,  AND  THE  LAWS  OF  MOTION. 

Attraction  defined,  .  .  .  .19 
Aggregation,  affinity,  cohesion,  .  19 

Gravity, 19 

Rapidity  of  its  communication,  .  19 
Its  relation  to  quantity  of  matter,  .  19 
Relation  to  the  center  of  the  earth,  .  19 
Weight  defined,  .  .  .  •  .  20 
Law  of  gravity  at  different  distan- 
ces,   20 

Variations  of  weight,  .  .  .21 
Case  of  a  body  within  a  hollow 

sphere, 22 

Forces  of  gravity  within  the  earth,  .  22 
Inertia  defined,  .  .  .  .23 
Its  force  as  the  quantity  of  matter,  .  23 
Uniform  velocity,  defined,  .  .  24 
Space  described  in  uniform  motions,  24 
Space,  time,  and  velocity,  how  re- 
lated,   24 

Momentum  defined,  .  .  .  .25 
Its  relation  to  quantity  of  matter,  .  26 
Relation  to  velocity,  .  .  .26 
Force  defined,  .  .  .  .27 

Forces,  impulsive,  accelerating,  con-     27 
stant,  aud  variable,        .         .         .27 


Velocities  generated  by  different  con- 
stant forces,  .         .         .         .27 
First  Law  of  Motion  stated,     .         .    27 

How  proved, 27 

Second  Law, 28 

How  proved, 28 

Third  Law, 29 

How  proved,  .  .  .  .  .29 
Extensive  applications  of  this  Law,  30 
Threefold  evidence  of  these  Laws,  .  30 
Observation  and  Experiment  defined,  30 

FALLING   BODIES. 

In  uniform  motion,  if  one  side  of  a 
parallelogram  represents  the  veloci- 
ty, and  the  other  the  time,  what 
represents  the  space  ?     .        .        .32 
Propriety  of  representing  the  space 

described  by  a  superficies,      .         .    33 
In  falling  bodies,  how  are  the  spaces 

to  the  times  ?         .  .    35 

Velocity  and  Time,  how  related,  .  35 
Spaces  described  in  equal  successive 

portions  of  time 35 

Ascending  bodies,  how  retarded,  .  36 
Space  described  by  the  last  acquired 

velocity,         ....  36 
Ditto,  in  an  ascending  body,      .             37 
Ditto,   by   a  body  projected   down- 
ward,      37 

Ditto,  by  a  body  projected  upward,  .  37 
Space  described  from  rest  in  1",  .  38 
Velocity  acquired  in  falling  1'',  .  38 

Case  of  a  body  falling  from  a  great 

distance,  .         .         .         .43 

Ditto,  when  projected  with  a  veloci- 
ty of  seven  miles  per  second,  .  44 

COMPOSITION    AND    RESOLUTION   OF 
MOTION. 

Motion  of  a  body  by  two  impulses, 
represented  by  the  two  sidea  of  a 
parallelogram,  ....  44 


ANALYSIS. 


Motion  of  a  body  by  two  impulses, 
represented  by  the  sides  of  a  tri- 
angle,   ......    46 

Ditto,  by  all  the  sides  of  a  polygon 

.    except  the  last 47 

Equivalent  force  defined,  .  .  .47 
Components  and  Resultant,  .  .  48 
Path  of  a  body  moved  by  a  constant 

and  a  variable  force,      .         .         .48 
Ditto,  by  a  projectile  force  and  grav- 
ity,   49 

Resolution  of  a  given  force,  .  .  51 
Ditto,  when  the  forces  act  at  right 

angles, 51 

When  they  act  at  any  given  angle,  .  52 
When  the  sum.  of  the  resolved  forces 

equals  a  given  quantity,  .  •  52 
When  the  difference  equals  a  given 

quantity,       .         .  .         .52 

Relative  magnitude  of  forces  before 

and  after  composition,  .  .  .53 
Ditto,  after  resolution,  .  .  .54 
Case  of  a  body  kept  at  rest  by  three 

forces, 55 

Any  of  the  three  forces  as  the  sine 

of  what  angle  ?  .  .  .  .56 
Case  of  a  body  kept  at  rest  by  any 

number  of  forces,  ....     56 
Parallel  forces,  direction  of  the  re- 
sultant,   58 

Components,  how  represented,  .     58 

Forces  acting  in  different  planes — all 
the  forces  that  can  act  upon  a 
body,  how  resolved,  .  .  .59 

CENTER    OF   GRAVITY. 

Center  of  Gravity  denned,  .  .  61 
In  plane  figures,  .  .  .  .62 
Pressure  on  a  prop  the  same  as  if  all 

the  matter  were  at  the  center  of 

gravity, 62 

When  are  two  bodies  in  equilibrium 

about  their  center  of  gravity  ?  .  63 
Product  of  the  two  bodies  into  their 

respective  distances,  .  .  .65 
How  to  find  the  common  center  of 

gravity  of  any  number  of  bodies,  .  65 
Line  of  direction  defined,  .  .  66 
When  will  a  body  be  at  rest  on  a 

horizontal  plane  ?  ...  66 

When  will  a  suspended  body  be  at 

rest? 66 

Center  of  gravity  of  a  triangle,  how 

situated, 67 

In  what  line  in  a  trapezoid,  .  .  67 
How  found  in  a  polygon,  .  .  68 

Distance  of  the  common  center  of 

gravity  of  any  number  of  bodies 

from  an  assumed  point,  .  .  69 
Case  of  two  bodies  movkg  from  and 

toward  each  other,  .         .     72 


Case  of  one  body  describing  any  fig- 
ure around  another  at  rest,  .  .  72 

Motion  of  the  center  of  gravity  of  a 
system, 73 

When  the  bodies  move  uniformly  in 
a  right  line 75 

How  affected  by  the  mutual  actions 
of  bodies,  .  .  .  ...  75 

If  the  motion  of  a  system  of  bodies 
in  their  orbits  should  cease,  how 
would  they  be  affected  ?  .75 

Distance  of  the  common  center  of 
gravity  of  any  number  of  bodies 
from  a  given  plane,  .  .  .78 

COLLISION    OF   BODIES. 

Elastic  bodies  defined,       .         .         .79 
Experiments,  how  made,  .         .         .79 
Case  when  one  inelastic  body  strikes 
on  another  at  rest,  or  moving  with 
less  velocity  in  the  same  direction,     80 
Velocity  lost  by  A,    .         .        .         .80 
Ditto,  gained  by  B,  .         .         .         .80 
Case  when  the  bodies  move  in  oppo- 
site directions,       .         .         .         .80 
When  two  bodies  meet  with  veloci- 
ties inversely  as  the  quantities   of 

matter, 80 

Elastic  bodies — velocity  gained  and 
lost  compared  with  inelastic  bod- 
ies,   81 

When  the  bodies  are  equal,  motion 

after  impact,          .         .         .         .82 
When   one  strikes   on  the  other  at 

rest, 83 

Case  of  a  row  of  equal  elastic  balls, 
motion  being  communicated  by  the 
first,  .  .  .  .  .83 

When  they  decrease  in  magnitude,  .     83 
When  they  increase,         .        ,        .83 
When   a  perfectly  elastic  body  im- 
pinges    on     a    perfectly    smooth 
plane, 84 

THE    LEVER. 

Mechanical  Powers  enumerated,      .  86 
Machines,  simple  and  compound,      .  86 
Lever  defined,  .        .         .         .         .86 
Fulcrum,  power,  weight,            .         .  86- 
Condition  of  equilibrium  of  two  forces 
acting  perpendicularly   at  the  ex- 
tremities of  a  straight  lever,  .         .  87 
How  the  effect  of  any  force  is  meas- 
ured,       87 

Condition  of  equilibrium  among  any 

number  of  weights,         .        .         .88 

Three  kinds  of  levers,        .         .        .  88 
Law  of  equilibrium  in  every  kind  of 

lever 90 

In  the  compound  lever,                       .  93 


To  find  the  weight  of  a  body  by  a 
false  balance 94 


WHEEL   AND    AXLE. 


Law  of  equilibrium,  .         .         .         .99 
Ditto  in  a  combination  of  wheels,      .  101 


Definition, 103 

Fixed  pulley 103 

Pressure  on  the  axis,          .         .         .103 
Use  of  the  fixed  pulley,     .         .        .103 
Law  of  equilibrium  in  the  movable 
pulley, 104 

INCLINED    PLANE. 

Law  of  equilibrium,  .  .  .  108 

When  the  power  acts  parallel  to  the 

plane, 108 

Ratio  of  the  power  to  the  weight,  .  108 

"        power  to  the  pressure  on  the 

plane, 108 

Ratio  of  the  weight  to  the  pressure,  .  108 
When  the  power  acts  parallel  to  the 

base  of  the  plane,  .  .  .109 
Direction  of  the  power  to  have  the 

greatest  efficacy,  .  .  .  .109 
Direction  to  produce  the  greatest 

pressure,  ...  109 

Equilibrium  between  two  weights 

across  a  ridge,  .  .  .  .112 
Comparative  pressure  when  a  weight 

is  sustained  between  two  inclined 

planes, 113 


Screw  defined, 
Law  of  equilibrium,  . 


.  113 
.  115 


Law  of  equilibrium,  .  .  .119 

Do.  when  the  directions  of  the  power 

and  resistances  meet  in  the  same 

point 120 

Do.  in  the  isosceles  wedge,  .  .  121 
General  principle  applicable  to  all 

the  mechanical  powers,  .  .  122 
Rope  Machine,  .  .  .  .  124 
Law  of  equilibrium,  ....  124 
Force  required  to  draw  a  rope  into  a 

straight  line  between  two  points,    .  125 

MOTION  ON  INCLINED  PLANES. 

Theorems, 128 

Velocity  acquired  Ji  falling  down  an 
inclined  plane,       .        .        .         .128 


Time  of  describiig  the  plane,    .         .  128 

Ratio  of  the  time  to  the  length,        .  128 

time  to  the  height,  128 

"  velocity  to  the  height,  128 

Times  of  describing  different  chords,  132 

Velocities  acquired,  ....  132 
Velocity  acquired  in  descending  from 

a  given  height  by  any  path,  .  .  133 
Ratios  of  times  and  velocities  to  the 

lengths  and  heights,  .  .  .  133 
Ratios  of  times  down  similar  systems 

of  planes  to  the  lengths,  .  .  134 

Do.  down  similar  curves,  .  .  134 

„  PENDULUMS. 

Definition, 135 

Center  of  suspension,  .  .  .135 
Vibration  of  a  pendulum,  .  .  .  135 
Center  of  oscillation,  .  .  .135 
Its  relation  to  the  center  of  gravity,  135 
Cycloid  defined,  .  .  .  .136 
Relation  of  the  cycloidal  ordinate  to 

the  corresponding  circular  arc,       .  136 
Relation  of  the  tangent  to  the  circu- 
lar chord, 136 

Relation  of  the  cycloidal  arc  to  the 

corresponding  circular  chord,  .  137 
How  to  make  a  pendulum  vibrate  in 

a  cycloid, 139 

Time  of  vibration  compared  with  the 
time  of  a  body's   falling  through 
half  the  length,     .         .         .        .139 
Times  of  vibration  in  pendulums  of 

the  same  length,  ....  140 
Times  of  pendulums  of  different 

lengths, 140 

Times,  when  the  lengths  and  forces 

are  both  different,  .         .         .140 

Number  of  vibrations  of  such  pendu- 

.     lums, 141 

Number  when  the  lengths  are  given,  141 
Number  when  the  forces  are  given,  141 
Times  of  vibration  of  the  same  pen- 
dulum with  different  forces,  .         .  141 

CENTRAL   FORCES. 

Central  Force  defined,       .         .         .144 
When  a  body  revolves  in  a  circle,     .  144 
Central  force,  how  related  to  the  ve- 
locity when  a  body  moves  uniform- 
ly in  a  circle,         ....  144 
How  related  to  the  time,  .         .         .144 

PROJECTILES. 

Definition 145 

Path  of  a  projectile,  .         .         .  145 

Investigation  of  the  general  formula,  145 

Theorems, 149 

Time  of  flight,  how  it  varies,   .        .  149 


Range, 1*49 

Greatest  height,  .  .  .  .149 
What  elevation  gives  the  greatest 

range, 149 

Elevation  to  give  the  greatest  height,  149 
Elevation  to  give  the  greatest  time 

of  flight, 149 

STRENGTH   OF   MATERIALS. 

Importance  of  these  inquiries  to  the 

architect  and  engineer,  .        .  151 

Strength  defined,      .        .         .        .151 

Stress,      .        .        .         .         .         .151 

Strength  of  a  beam  resting  horizon- 
tally on  its  two  ends,     .         .         .151 
In  cylindrical  and  square  beams,       .  152 
In  beams  of  different  lengths  on  two 
supports,       .         .  .         .153 

Triangular  beams,    .         .         .         .153 

Strength  of  any  bar  in  the  direction 
of  its  length,          .         .         .        .153 

Lateral  strengths  of  similar  beams,    .  154 
Weakness  if  long  beams,          .         .  154 
Weight  sustained  by  a  beam  of  oak 
1  foot  long  and  1  inch  square,        .  155 

Ditto  of  iron 155 

Stress  on  a  horizontal  beam  support- 
ed at  both  ends,    .         .         .         .156 
Weakest  point  of  the  beam,       .         .  156 
Form    of    a    beam    equally    strong 

throughout,  .         .         .         .         .157 
Application  of  this  principle,     .         .  157 
Strength  of  similar  cylindrical  and 
prismatic  beams  supported  atone 

end 159 

Form  of  such  a  beam  when  equally 

strong  throughout,          .        .         .160 
How  much  stronger  is  a  beam  when 
supported  at  both  ends  than  at  one 

end? 162 

Comparative  strength   of  great  and 

small  beams,  ....  163 
Magnitude  of  natural  objects  limited,  163 
Comparative  strength  of  very  large 

animals 163 

Lateral  strengths  of  two  cylinders  of 

the  same  matter 164 

How    nature   increases  the  strength 
of  substances 164 

MECHANICAL   PROPERTIES   OF   MATTER. 

Matter,  how  related  to  Chemistry,  .  166 
Motion,  how  related  to  Mechanical 

Philosophy 166 

Of  what  sort  of  matter  it  takes  cogni- 
zance,    166 

Leading  Properties  of  Matter,  .  .  166 
Divisibility — illustrations,  .  .166 
Porosity,  do.  .  .  .167 

Compressibility,     do.         .         .        .167 


Elasticity — illustrations,    .         .         .  168 
Indestructibility,    do.         .         .         .169 
Attraction— extent  of  its  action,        .  168 
Strength  of  substances  in  the  direc- 
tion of  their  lengths,  how  ascer- 
tained,   169 

Examples, 169 

Force  required  by  different  substan- 
ces to  crush  them,         .         .         .169 
What  constitutes  the  strength  of  a 

beam  according  to  Barlow,    .         .170 
Comparative  strength  of  wood  and 
iron,      .......  171 

Trees,  then-  mechanical  structure,     .  171 

GENERAL   OBSERVATIONS    ON   MOTION. 
I 

Absolute  and  Relative  motion,           .  172 
Apparent  and  Real,  .         .         .         .172 
First  Law,  Inertia  in  four  particu- 
lars  173 

Distinguished  from  weight,        .         .  173 
"  from  reaction,      .         .173 

Communication  of  motion  to  the  en- 
tire mass  gradual,          .         .         .  174 
Examples  of  a  natural  tendency  to 

uniform  motion 176 

Atwood's  Machine  described,  .  .176 
Tendency  to  move  in  right  lines,  .  177 
Centrifugal  force,  ....  177 
Relation  of  this  force  to  the  densities 

of  bodies, 178 

Ditto  to  the  velocities,  .  .  .178 
Effect  on  the  figure  of  a  revolving 

sphere, 178 

Effect  on  the  weight  of  bodies,  .  179 
How  the  doctrine  of  central  forces  is 

extended  to  all  curves,  .  .  .179 
Second  Law  of  motion,  .  .  .  180 
That  it  is  proportioned  to  the  force 

impressed,  how  proved,  .         .  180 

That  it  is  the  direction  of  the  force 

impressed,  .....  180 
Whether  the  smallest  force  can  move 

the  largest  body 181 

Third  Law  of  motion — illustrations,  181 
Sum  of  the  motions  of  all  bodies,  es- 
timated in  a  given  direction,           .  183 
Two  methods  of  obtaining  great  mo- 
mentum,         183 

Blows  received  by  ships  when  they 

meet 184 

Comparative  effects  of  a  fall  on  hard 

and  on  soft  bodies,         .        .        .184 
Cases  where  action  and  reaction  de- 
stroy each  other,  ....  184 
Variable  motion  defined,  .         .         .185 

Examples, 185 

Uniformly  accelerated  motion,  .  185 

Force  of  gravity  at  the  distance  of 

the  moon, -186 

Effect  of  gravity  on  projectiles,          .  186 


ANALYSIS. 


Page. 

Velocity  of  a  projectile  that  would  go 
round  the  earth,  ....  187 

Gravity  manifested  by  small  masses 
of  matter, 187 

Why  gravity  is  directed  toward  the 
center  of  the  earth,  .  .  .  187 

Difficulty  of  verifying  the  law  of 
gravity  by  experiment,  .  .  188 

Means  by  which  these  difficulties 
are  surmounted  in  Atwood's  Ma- 
chine,   188 

How  far  a  weight  falls  in  one  sec- 
ond,   189 

Do.  in  six  seconds  compared  with  a 
body  falling  freely,  .  .  .189 

Examples  of  accelerated  motion  in 
nature, 189 

Slide  of  Alpnach,      .         .         .         .190 

Composition  and  Resolution  of  Mo- 
tion, .  .  .  .  .190 

Simple  and  compound  motion,  .  190 

Law  of  the  composition  of  motion, 
how  illustrated  by  experiment,  .  191 

Examples,  a  ship,  a  kite,  .         .  193 

CENTER   OF   GRAVITY. 

Simplicity  attained  by  the  doctrine,  .  196 
The  resultant  of  -parallel  forces,  .  197 
To  find  it  by  experiment,  .  .  197 

Properties, 198 

Stable  position  of  a  globe  on  a  small 

base, 198 

Two  points  of  rest  in  a  body  revolv- 
ing in  a  vertical  plane,  .         .  198 
Stability  of  a  body,  how  related  to 

the  line  of  direction,      .         .         .199 
Leaning  towers,  Rocking  stones,       .  199 
Motions  of  animals,  Stability  of  trees,  200 
Position  of  the  center  of  gravity  not 
altered   by  the   mutual   action  of 

bodies, 200 

Problems   solved    by  means  of   the 

center  of  gravity,  .         .         .  201 

Surface  generated  by   any  line  re- 
volving about  a  fixed  point,    .         .  202 
Solid  generated  by  the  revolution  of 
any  surface,          .        .        .        .202 

MACHINERY. 

Tools,  machines,  and  engines  distin- 
guished,        .....  203 
State  of  the  mechanic  arts  among  the 

ancients, 203 

Lever,  its  extensive  applications,  .  204 
Balance,  its  construction,  .  .  204 
Construction  of  a  perfect  balance,  .  204 
Bent  lever  balance,  ....  206 

False  balance, 206 

Steelyard, 206 

Spring  steelyard,       .         .        .         .207 


Weighing  machine,  ....  207 
Mode  of  estimating  the  weight  of  a 

body  too  heavy  for  the  steelyards,  208 
Examples  of  levers  of  the  first  kind,  208 
Levers  of  the  second  kind,  .  .  208 
Levers  of  the  third  kind, .  .  .209 
Bones  of  animals,  .  .  .  .209 
Advantages  and  disadvantages  of  this 

arrangement,          ....  210 

WHEEL  WORK. 

Use  for  continued  motion,  .  .  210 
Structure  of  the  compound  axle,  .  211 
Rationale  of  its  action,  .  .  .212 
Principle  of  watch  springs,  .  .  212 
Methods  of  communicating  motion 

by  wheel  work,  ....  213 
Regulation  of  velocity  by  wheel  work,  &15 
Example  in  the  lathe,  .  .  .215 
In  clock  work,  .  .  .  .216 

Relation  of  the  velocity  of  the  wheel 

and  pinion  to  the  number  of  teeth,  216 
What  maintains  the  motion  of  the 

pendulum, 217 

Wheel  Carriages,  .  ."  .  .217 
In  what  the  resistance  to  the  hori- 
zontal motion  of  a  load  consists,  .  217 
Advantage  gained  by  transferring  the 

friction  from  the  ground  to  the  axle,  218 
Advantages  of  wheels  in  overcoming 

obstacles, 218 

Causes  of  the  heavy  action  of  wheels 

in  a  yielding  soil,  ....  219 
Direction  of  the  line  of  draught,  .  219 
Effect  of  springs,  .  .  .  .219 
Most  advantageous  position  of  horses,  220 

Pulley, 220 

Fixed  and  movable  pulleys,  .  .  220 
Uses  of  the  fixed  pulley,  .  .  .  22ft 

Fire  escapes 221 

Movable  pulleys,  their  mechanical 

advantage, 221 

Inclined  Plane,  .  .  .  .222 
How  it  becomes  a  mechanical  power,  222 
Examples  of  its  application,  .  .  222 
Railways,  their  construction,  .  .  223 
Canals,  when  more  advantageous 

than  railways,  ....  224 

Screw, 224 

When  is  the  screw  employed  ?  .  .224 
Endless  screw,  .  .  .  .  224 
Hunter's  screw,  construction  and 

principle, 225 

Micrometer  Screw,  construction  and 

use, 226 

Wedge 226 

Use  in  the  arts,  .  .  .  .227 
How  its  power  is  increased,  .  .  227 
Friction  of  the  wedge,  its  use,  .  .  227 
Advantages  of  Machinery,  .  .  228 
Boast  of  Archimedes,  .  .  .228 


Machines  of  two  classes,  .  .  .228 
How  the  effect  is  measured  in  each, .  228 
No  momentum  gained,  .  .  .  229 
Uses  of  machinery  enumerated,  .  229 

REGULATION  OF  MACHINERY. 

Utility  of  regularity  and  uniformity  in 

machinery 231 

Regulators,  their  office,     .         .         .232 
How  large  machines  regulate  them- 
selves,     232 

Fly  wheel,  principle  and  use,  .  .  232 
Use  to  accumulate  power,  .  .  233 
Principle  of  the  sling,  .  .  .233 
No  increase  of  force  by  the  fly,  .  234 
Use  of  balls  to  accumulate  power  in 

coining  presses,     ....  234 
Governor,  its  construction  and  prin- 
ciple,      235 

CONTRIVANCES   FOR   MODIFYING    MOTION. 

Different  species  of  motion  required 

in  different  cases 236 

Simplest  mode  of  changing  the  direc- 
tion  236 

Gearing,  different  kinds,  .  .  .237 
Universal  joint,  ....  237 
Ratchet  wheel,  .  .  .  .237 
Eccentric  wheel,  ....  237 
Crank  motion  described,  .  .  .  238 

Arch  head, 238 

Knee  joint, 239 

Rules  in  the  choice  and  use  of  ma- 
chines  240 


Resistance  from  cohesion,  .  .  241 
Ditto,  from  inequalities  of  surface,  .  241 
Passive  forces,  their  nature,  .  .  242 
Experiments  of  Coulomb  and  Vince,  242 
Two  forms  of  friction,  sliding  and 

'    rolling, 242 

Mode  of  conducting  experiments,  .  242 
Extent  of  surface,  its  influence,  .  243 

Pressure, 243 

Continued  contact,  ....  243 
Friction  between  different  kinds  of 

matter 243 

Amount  at  first  moving  a  load,  .  244 
How  affected  by  different  velocities,  .  244 
Observations  at  the  slide  of  Alpnach,  244 
Angle  of  friction  or  repose,  .  .  245 
Comparative  amount  in  sliding,  re- 
volving, and  rolling  bodies,  .  .  245 
Friction  wheels,  construction  and 

principle 245 

Methods  of  diminishing  friction,         .  246 
Comparative  friction  of  the  different 
mechanical  powers,       .  .  246 


Utility  of  friction,     .        .        .  246 

PROJECTILES  AND   GUNNERY 

Views  of  Galileo,  .  .  .  .247 
Experiments  of  Robins,  .  .  .247 
Curve  described  by  bodies  moving 

slowly, 248 

Ditto,  moving  swiftly,       ,        .        .  248 
Discordance  between  theory  and  prac- 
tice in  gunnery 248 

Ballistic  pendulum,  .        .         .249 

Velocity  of  a  cannon  ball,  .  .  250 
Great  charges  of  powder  useless,  .  251 
Ratio  of  the  charge  to  the  ball,  .  251 
Velocity  with  which  the  charge  ex- 
pands, .  .  ' .  .  .  .251 
Carronades,  .  .  .  .  .  ,552 
Windage,  its  effects,  .  .  .252 
Rifles,  their  theory,  .  .  .  .252 
Ricochet  firing,  .  .  .  .252 

APPLICATIONS   OF  THE    PENDULUM. 

Three  distinct  applications,        .         .  253 

Time  defined, 253 

Measures  of  time,     .         .         .         .253 
History  of  the  application  of  the  pen- 
dulum to  this  use,          .         .         .  253 
Different  lengths  employed,       .  254 

Cycloidal  pendulum,          .         .  254 

Sources  of  inaccuracy,  .  .  .  255 
Compensation  pendulums,  .  .  255 
Gridiron  pendulum  described,  .  .  256 
Use  of  the  pendulum  in  investigating 

the  figure  of  the  earth,  .         .256 

Ellipticity  of  the  earth,     .         .         .  257 
Use  of  the  pendulum  as  a  standard 
of  measures,  ....  257 

Ancient  standards 258 

Efforts  of  the  French  Government,  .  258 
Metre,  its  length,  .  .  .  .258 
Standard  adopted  in  Great  Britain 

and  New  York,     .         .         .         .258 
Regulation  of  weights  and  measures 

in  New  York 258 

Do.  by  the  government  of  the  United 
States, 259 

HYDROSTATICS. 

Meaning  of  the  terms  hydrostatics, 
hydraulics,     hydrodynamics,    and 

pneumatics 260 

Use  of  experiments  on  this  subject,  .  260 

Fluid  defined 260 

Elasticity  of  water 261 

Experiment  of  the  Florentines,          .  261 
Ditto,  of  Perkins,     .         .         .         .261 
Contraction  from  a  pressure  of  2000 
atmospheres,          ....  261 


ANALYSIS. 


FLUIDS   AT    REST. 

Hydrostatics  defined,  .  .  .  262 
Equal  pressure  of  the  parts  of  a  fluid,  262 
Effect  of  a  blow  or  pressure  on  any 

part  of  a  confined  fluid,          .         .  262 
Hydrostatic  press  described,      .         .  263 
Force  which  a  man  can  exert  by  it,    263 
Principle  of  this  press,       .         .         .  264 
Its  uses,    ......  264 

'Fluids  at  rest  horizontal,  .  .  .264 
'Curvature  for  100  feet,  .  .  .265 
Levelling,  how  practised,  .  .  266 

Spirit  level  described,  .  .  .  266 
Pressure,  how  related  to  depth,  .  266 
Actual  pressure  on  a  square  foot  at 

various  depths,  ....  267 
Effects  on  whales  .....  268 
Ditto,  on  bottles,  .  .  .  .268 
Compression  of  water  at  the  depth  of 

one  mile,  .....  268 
Effects  of  the  pressure  at  great  depths 

on  sunken  ships,  ....  268 
Pressure  against  any  surface,  .  .  269 
Level  of  a  fluid  in  opposite  arms  of  a 

recurved  tube,  ....  269 
Ditto,  in  vessels  of  various  shapes  all 

connected  with  the  same  reservoir,  270 
Height  to  which  water  will  rise  in 

aqueducts,  .....  270 
Aqueducts  of  the  ancients,  .  .  270 
Pressure  upon  a  horizontal  base,  to 

what  equal  ......  271 

Hydrostatic  paradox,         .        .         .271 
Explanation,      .....  271 

Why  the  weight  of  a  given  quantity 

of  water  is  the  same  in  different 

shaped  vessels,       .         .         .         .272 

SPECIFIC   GRAVITY. 

Definition,  .....  272 
272 
272 
272 
273 


Standard  for  solids  and  liquids, 
Ditto,  for  gases,  .  .  . 
Loss  of  weight  in  water,  .  . 
Rule  for  specific  gravity,  . 

Ditto,  when  the  body  is  lighter  than 

water,  .....  273 

Ditto,  when  the  body  is  soluble  in 

water,    ......  273 

To  find  the  specific  gravity  of  liquids,  273 
Equilibrium  of  different  fluids  in  a 

recurved  tube,       ....  374 
Precautions  where  great  accuracy  is 

required,        .....  274 
Floating   bodies,   how  much   water 

they  displace,         ....  274 
Ratio  of  the  part  immersed  to  the 

whole  .......  275 

Position  of  the  center  of  gravity  of 

floating  bodies  .....  275 
Attempts  to  walk  on  water,      .  276 


Life-preservers  described,  276 

Metacenter,  .         .  276 

Common  hydrometer,  .  .  .  277 
Nicholson's  ditto,  ....  278 
How  to  determine  by  it  the  specific 

gravity  of  a  solid,  .         .         .  278 

Utility   of   the   doctrine   of    specific 

gravities, 279 

Specific  gravity  of  platinum  and  hy- 
drogen compared,  .         .         .  279 
Specific  gravity  of  metals,  ores,  gems, 
stones,   acids,  oils,   milk,   alcohol, 

woods, 279 

Case  of  a  ship  floated  with  a  small 

quantity  of  water,  .  .  .  280 
Weight  of  the  ship,  how  estimated,  280 
Specific  gravity  of  the  human  body,  280 
Force  with  which  a  body  ascends  or 

descends  in  a  fluid,  .  .  .281 
Camel,  its  construction  and  principle,  281 
Rocks  elevated  by  ice,  .  .  .281 

Life-boats 281 

Magnitude   of   bodies   estimated  on 
the  principle  of  specific  gravity,        282 

LIQUIDS   IN   MOTION. 

Hydraulics  defined,  .  .285 

Subjects  of  which  it  treats,  .  .  285 
Theory  discordant  with  facts,  .  .  285 
Manner  in  which  fluids  discharge 

themselves  from  a  small  orifice,  286 
Vena  contracta, ,  .  .  .286 

Velocity  of  a  fluid  in  different  parts 

of  a  tube  of  unequal  bore.     .         .  286 
Mean  Velocity  of  a  stream,  how  de- 
termined,        28t> 

Effect    of  expansions   and   contrac- 
tions on  the  velocity  of  a  stream,     286 
To  find  the  quantity  of  water  which 

flows  in  a  river 287 

Rivers,   whether    their    phenomena 

can  be  explained  according  to  the 

laws   of  bodies   falling   down   in- 

clined  planes,         ....  287 

Cause  of  the  increased  velocity  in  a 

freshet, 287 

Why  rivers  run  so  slowly,  .  .  287 
Smallest  inclination  capable  of  giving 

motion  to  water,  ....  287 
Velocity  as  the  square  root  of  the 


Advantage  of  taking  water  from  the 

bottom  of  a  reservoir,  .  .  .  288 
Rate  at  which  the  surface  descends 

in  a  cylindric  or  prismatic  vessel,  289 
Construction  of  the  Clepsydra,  .  .  289 
Increased  amount  discharged  when 

the  vessel  is  kept  full,  .  .  .289 
Curve  described  by  a  spouting  fluid,  290 
Random  of  a  spouting  fluid,  .  290 

Greatest  random,      .        .  290 


ANALYSIS. 


Barker's  mill, 291 

Friction,  extent  of  its  influence,  .  292 
How  obviated,  .  .  .  .292 

Effect  of  suddenly  stopping  a  large 

body  of  running  water,  .  .  292 
Effect  of  a  pipe  applied  to  the  end  of 

a  discharging  vessel,  .  .  .  293 
Length  of  the  delivering  pipe,  .  .  293 
Its  shape, 293 

WATER   WHEELS. 

Three  kinds, 293 

Overshot  wheel,  when  used,      .         .  294 

Description, 294 

Best  velocity, 294 

Undershot  wheel,      .        .        .        .295 

How  carried,     .....  295 

Description,       .        ....        .295 

Use  in  tide  mills,       .        .         .        .295 

Breast  wheel,  when  used,  .  .  295 
Description, 295 

CAPILLARY  ATTRACTION. 

Cohesion  of  fluids,  .  .  .  .296 
Adhesion  of  disks  to  the  surface  of  a 

fluid, 296 

Capillary  attraction  defined,  .  .  297 
Size  of  the  tubes,  .  .  .  .297 
Height  to  which  fluids  will  rise  in 

tubes  of  different  bores,  .  .297 
Different  fluids  raised  to  unequal 

heights, 297 

Height  due  to  a  tube  _£._  inch  diam- 
eter,     .        .        /  °    .        .        .297 
Product   of    the   diameter    into   the 

height 297 

Rise  of  fluids  between  plates,     .         .  297 
Effect  when  a  capillary  tube  is  im- 
mersed in  mercury,        .         .         .297 
Case  when  the  tube  is  wider  at  bot- 
tom than  at  top,     .         .         .         .297 
Cause  of  capillary  action,          .         .  298 
Natural  phenomena  explained,  .  298 

Examples  of  capillary  action,    .         .  298 

KESISTANCE  OP  FLUIDS. 

r  Causes  of  this  resistance,  .  .  .  299 
Law  of  resistance,  ....  299 
How  modified  in  great  velocities,  .  300 

FORMATION  OF   WAVES. 

How  formed, 301 

Depth  to  which  the  winds  penetrate,  302 
Why  waves  run  so  high  in  a  storm,  302 
Spray  and  breakers  defined,  .  .  302 


PNEUMATICS. 

Pneumatics  defined,          . 


.  303 


Laws  common  to  hydrostatics  and 

pneumatics,  ....  303 

Vapors  and  gases  distinguished,  .  303 
Heat — general  law  of  expansion,  .  303 
Properties  of  air,  .  .  .  .304 
Its  materiality,  how  proved,  .  .  304 
Its  fluidity  and  elasticity,  .  304 

Law  of  Mariotte,  .  .  .  .305 
Air  Pump  described,  .  .  .306 
Valve  defined,  .  .  306 

Piston  and  Cylinder  described,  .  306 

Process  of  exhaustion,       .         .         .  308 

Rate  of  ditto, 308 

Exhaustion,  why  imperfect,  .  .  309 
Pressure  of  the  atmosphere,  per 

square  inch,  .         .         .         .309 

Its  amount  on  the  human  body,  .  309 
Present  in  the  pores  of  bodies,  .  .  310 
Its  effect  on  the  boiling  point  of  liquids,  310 
Degree  of  its  elasticity,  .  .  .310 
Its  relatioji  to  sound,  respiration,  and 

combustion 310 

Its  effect  on  the  weight  of  bodies,  .  310 
Condensing  Syringe  described,  .  310 

Condensing  Fountain,  .  .  .311 
Geysers  of  Iceland,  .  .  .  .311 

Air  Gun, 311 

Diving  Bell, 311 

How  supplied  with  fresh  air,  .  .  312 
Barometer  described,  .  .  .  312 
Torricellian  Vacuum,  .  .  .312 
How  the  mercurial  column  shows  the 

pressure  of  the  atmosphere,  .         .  313 
How  to  estimate  the  weight  of  the  at- 
mosphere from  the  barometer,       .  313 
Mode  of  graduating  it,     .         .         .  314 

Vernier  described 314 

Portable  or  mountain  barometer,        .  314 
Indications   of  weather  by  the  ba- 
rometer,          315 

State  of  the  barometer  before  a  gale 

of  wind, 315 

Amount    of   pressure   in    different 

places, 315 

Horary  variations,     ....  316 

Height   above  the  earth  correspond- 
ing to  a  descent  of  one-tenth  of  an 
inch,     .         .         .         .         .         .  316 

How  much  heavier  are  mercury  and 
water  than  air,      .         .         .         .316 

Gauge  described,       .         .         .         .316 

Ratio  of  the  elasticity  to  the  temper- 
ature,  317 

Effect  of  continued  pressure  on  the 

elasticity  of  the  air,  .  .  .317 
Weight  of  the  entire  atmosphere,  .317 
How  estimated,  ....  317 
Weight  if  the  d.ensity  were  uniform,  318 
Causes  which  prevent  its  uniformity,  318 
Law  of  density  at  different  heights,  319 
Density  at  the  height  of  21,  35,  and 
49  miles, 319 


Actual  height  of  the  atmosphere,  .  320 
Proofs  that  it  comes  to  definite  limits,  321 
Cold  of  the  upper  regions — term  of 

perpetual  congelation,  .  .  .  321 
Its  height  at  the  equator,  at  Lat.  30°, 

40°,  54Q,  and  80°,         .         .         .322 
Rate  of  decrease  in  the  height  in  dif- 
ferent latitudes,      .         .        .         .322 
Causes  of  the  cold  in  the  upper  re- 
gion of  the  atmosphere,         .         .  322 

RELATIONS  OF    AIR   TO  HEAT  AND  MOISTURE. 

Mobility  of  air,  .  .  .  .323 
Effect  of  a  change  of  temperature,  323 
Exemplified  in  the  ventilation  of 

mines, 323 

Ascent  of  smoke  in  chimneys,  .  .  324 
Circumstances  affecting  the  draught, 

as  length  of  the  flue,  width  of  the 

throat,  and  height  of  the  breast,  .  325 
All  the  air  should  pass  through  the 

fire  before  it  enters  the  chimney,  .  326 
Principles  of  Olmsted's  stove,  .  .  326 
Winds — general  cause,  .  .  .  328 
Land  and  sea  breezes  explained,  .  329 
Trade  winds  described,  .  .  .329 
Ditto,  explained,  ....  329 
Moisture,  how  raised  into  the  atmo- 

r1  ere, 330 
n's  views  of  the  constitution  of 
the  atmosphere,     ....  330 
Dew,  how  formed,     .         .         .         .331 
How  deposited  on  different  substan- 
ces,         331 

Fogs,  how  produced,  .  .  .331 
Clouds,  how  formed,  .  .  .  332 
Rain,  how  produced,  .  .  .  332 
Its  relation  to  ste'ady  and  to  variable 

winds, 332 

Hail,  how  produced,  .  .  .  332 
Countries  where  hail  storms  are  most 

violent 333 

Redfield's  theory  of  hurricanes,  .  333 
Espy's  do 333 

MECHANICAL  AGENCIES   OF  AIH   AND  STEAM. 

How  air  and  steam  become  mechan- 
ical agents, 334 

Syphon  described  and  explained,  .  334 
Length,  how  limited,  .  .  .335 
Intermitting  springs  described  and 

explained,     ...  .  335 

Suction  Pump  described,  .  336 

Principle  enunciated,        .  .  336 

Effective  force  which  resists  the  as- 
cent of  the  piston,         .         .         .337 
Column  of  water  discharged  at  each 

stroke,  .         .         ;         .        .338 

Forcing  Pump  described,  .        .         .  338 
Principle  enunciated,         .        .         .  339 
2 


Fire  Engine  described,  .  .  .340 
Use  of  the  air  vessel,  .  .  .  341 
Hungarian  machine  described,  .  341 
Its  principle  enunciated,  .  .  .  342 
Steam  Engine,  .  .  .  .342 

Leading  properties  of  steam,  .  .  343 
Peculiar  property  on  which  its  ne- 

chanical  agencies  depend,  .  .  343 
On  what  does  the  elastic  force  depend,  343 
Elastic  force  of  steam  at  different 

temperatures,  ....  344 
Space  into  which  a  given  quantity  of 

water  expands  on  becoming  steam,  345 
Absolute  quantity  of  heat  the  same 

at  all  temperatures,  .  .  .  345 
History  of  the  Steam  Engine,  .  345 
Structure  of  the  Atmospheric  En- 
gine,   345 

Waste,  how  occasioned  by  it,  .  .  346 
Structure  and  principle  of  Watt's 

Condenser, 347 

Construction  and  use  of  the  air-pump,  348 
Advantage  of  using  steam  to  force 

down  the  piston,  ....  348 
Structure  of  the  Jacket,  .  .  348 

Saving  by  Watt's  contrivances,  348 

Description  of  the  steam  engine  in 

its  simplest  form,  ....  349 

Puppet  valve 350 

Advantage  of  cutting  off  the  steam, 

or  of  causing  it  to  act  expansively,  350 
Description  of  the  steam  engine  in 

its  complete  form,  .  .  .  351 

High  Pressure  Engines,  principle  and 

use, 352 

Invention  of  steam  boats,  .  .  353 

ACOUSTICS. 

Definition  of  Acoustics,  .  .  .  354 
Production  of  sound  in  general,  .  354 
Sonorous  qualities  of  different  bodies,  354 
How  they  differ  from  each  other  in 

nature  and  form,  ....  355 
Immediate  cause  of  musical  sounds,  355 
When  have  sounds  the  same  pitch,  .  355 
Musical  Strings — how  constructed,  356 
Stretching  force,  how  applied,  .  .  356 
Three  circumstances  on  which  the 

jitch  depends,  ....  356 
Vibrations  of  the  string  equal  in  equal 

times, 357 

Isochronism  in  vibrations  essential  to 

musical  sounds,  ....  357 
Frequency  of  vibration,  how  related 

to  the  length  of  the  string,  .  .  357 
Do.  to  the  weight,  ....  357 
Do.  to  the  tension,  .  .  .  358 

WIND    INSTRUMENTS. 

What  constitutes  the  vibrating  body,  358 


10 


ANALYSIS. 


Influence  of  the  material  of  the  in- 
strument  358 

Variety  in  bodies  vibrating  longitu- 
dinally  358 

Case  of  a  vibrating  body  fixed  at 
both  ends,  or  at  one  end,  .  .  358 

Pitch  of  bodies  vibrating  longitudi- 
nally, free  at  both  extremities,  .  358 

The  voice  and  the  ear,      .        .  358 

Bells — change  of  figure,  .         .        .  359 

PROPAGATION    OF   SOUND. 

Air  the  common  medium,  .  .360 
Sound  on  high  mountains,  .  .  360 
Sound  of  meteoric  bodies  at  great 

heights,  .  .  .  .  .360 
Sound  in  condensed  air,  .  .  .  360 
How  sound  is  communicated  from 

the  sounding  body  to  the  ear,         .  360 
Sound  propagated  in  all  directions,  .  361 
Law  of  its  intensity  at  different  dis- 
tances,   361 

Sound,  how  affected  by  an  intervening 

body, 361 

Whether  it  moves  in  right  lines,  .  362 
Various  conductors  of  sound,  .  .  362 
Velocity  of  sound,  .  .  .  .362 
Effect  of  heat  and  cold,  .  .  .362 
Velocity  uniform,  .  .  .  .363 
Experiment  of  Blot,  •  .  363 

Effect  of  the  wind,  .         .         .  363 

Experiments  made  in  Holland,  .  363 
Distance  of  the  sounding  body,  how 

estimated, 364 

Effect  of  humidity  of  the  air  upon  sound,  3  64 
Distance  to  which  sounds  are  audible,  364 
Conducting  power  of  liquids,  .  .  365 
Velocity  of  sound  under  water, .  .  365 
Conducting  power  of  solids,  .  .  366 
Stethoscopes-construction  and  use,  .  367 

REFLEXION    OF   SOUND. 

Law  of  reflexion,       ....  367 
Echo,  how  produced,         .        .        .368 
Estimate  of  distance  by  echo,  .         .  368 
-  How  the  furniture  of  a  room  affects 

sound 368 

Best  form  of  rooms,  ....  369 
Whispering  galleries,  .  .  .  369 
Rolling  of  thunder— cause,  .  .  370 
Speaking  Trumpet  explained,  .  .  371 

Ear  Trumpet, 371 

Acoustic  Tubes,  .  .  .  .371 
Ventriloquism  explained,  .  .  .371 
Sounding  Boards,  ....  372 
Sea  shells,  sonorous  qualities,  .  .  372 

PHILOSOPHICAL   TRINCIPLES   OF   MUSIC. 

How  sounds  become  musical,  .  37' 


Why  musical  sounds  fall  within  the 

province  of  Mathematics,       .        .  374 
Ratios  between  the  lengths  of  por- 
tions of  a  string  sounding  the  eight 
notes,     ...  .  374 

Three  kinds  of  intervals  in  the  mu- 
sical scale  ......  375 

Melody,  chords,  harmony  denned,  .  376 
Chords  —  whence  they  arise,  .  .  376 
Discords,  their  cause,  .  .  .  376 
Harmonics  explained,  .  .  .  377 
Theory  of  musical  instruments,  378 


ELECTRICITY. 


Etymology  of  the  name,    . 
History  of  the  science,      . 


.  380 
.380 


GENERAL   PRINCIPLES   OF   THE    SCIENCE. 

Attraction  the  most  general  effect,  .  382 
Definitions  —  excited,  electrified,  con. 

ductors  and  non-conductors,  elec- 

trics, insulated,  electroscopes  and 

electrometers,  ....  382 
Pendulum  Electrometer  described,  382 
Gold  Leaf  do.  ...  383 

Coulomb's  do.  ...  383 

Electricity,  how  produced,  .  383 

Other  methods  besides  friction,  .  383 
Whether  all  bodies  afford  electricity 

by  friction,  .....  384 
Opposite  properties  of  the  electricity 

of  glass  and  amber,  .  .  .  384 
Terms  positive  and  negative,  .  .  385 
Two  hypotheses  of  electricity,  .  .  385 
Law  of  electrical  attractions  and  re- 

pulsions, .....  385 
To  determine  the  kind  of  electricity,  385 
The  two  kinds  produced  simultane- 

ossly,    ......  386 

What  substances  afford  the  positive 

and  what  negative  electricity,  .  386 
Conductors  and  non-conductors,  spe- 

cified,   ......  386 

Effect  of  change  of  temperature  on 

conducting  power,  .  .  .  387 
Conductors  and  non-conductors,  .  387 
Conducting  power  of  Metals,  Water, 

and  Animals,  ....  388 
Do.  of  the  Earth  .....  388 
Do.  of  Air,  and  all  resinous  and  vit- 

reous substances,  .  .  .  .388 
Insulation  —  how  effected,  .  .  388 
Condition  of  the  air  essential  to  insu- 

lation,  ......  388 

Best  insulating  substances,        .        .  388 
Sphere  of  communication,         .        .  388 
Do.  of  influence,        ....  388 

Induction  defined  .....  389 

Wrhy  unelectrified  bodies  are  attracted,  389 
Return  stroke,  .  .  389 


ANALYSIS. 


11 


ELECTRICAL   APPARATUS. 

Object  of  electrical  apparatus,  .  .  390 
Cylinder  Machine  described,  .  .  390 
Amalgam,  how  formed,  .  .  .  391 
Plate  Machine  described, .  .  .391 
Quadrant  Electrometer,  .  .  .  392 
Construction  of  a  cheap  apparatus,  .  393 
Circles  of  light  on  turning  the  ma- 
chine explained,  ....  393 
Experiments  with  the  machine,  .  394 
How  bodies  are  electrified  positively,  394 
Do.  negatively,  .  .  .  .394 
Separation  of  bodies  into  minute  parts,  395 
Electrified  air  and  a  current  of  air,  .  395 
Torsion  Balance — inventor,  .  .  395 
Force  employed  as  a  measure,  .  .  395 
Description  of  the  instrument,  .  .  395 
How  the  forces  of  attraction  and  re- 
pulsion are  estimated,  .  .  .  395 
Law  of  attraction  and  repulsion  at 

different  distances,  .  .  .395 
Rate  of  dissipation  of  electricity,  .  398 
Distribution  of  electricity,  .  .  399 
Relation  to  the  surface,  .  .  .399 
Intensity  affected  by  extent  of  sur- 
face,   400 

Effect  of  the  figure  of  bodies  on  the 

distribution  of  electricity,       .         .  400 
Why  points  dissipate  so  freely,          .  401 

LEYDEN   JAR. 

Construction, 402 

Discharging  rod,  ....  402 
History  of  the  Leyden  Jar,  .  .  402 
Experiments  with  it,  '.  .  .  404 
How  charged  and  discharged,  .  404 

Opposite  sides  in  different  states,  404 
Outside  must  be  uninsulated,  .  .  404 
A  second  jar  charged  from  the  first,  404 
To  charge  a  jar  negatively,  .  .  404 
Case  of  two  jars  charged  differently,  405 
Electrical  Spider,  .  .  .  .405 
Division  of  a  charge  into  aliquot  parts,  405 
Office  of  the  coatings,  .  .  .  405 
Jar  retains  its  charge,  .  .  .  406 
Charging  a  pane  of  glass,  .  .  406 
Do.  a  plate  of  air,  .  .  .  .406 
Law  of  Induction,  ....  407 
Experiment  with  a  globe  and  cylinder,  407 
Whether  in  induction  there  is  any 

transfer  of  the  fluid,      .         .         .409 
Disguised  electricity,         .        .         .  410 
Why  bodies  differently  electrified  at- 
tract,      411 

Theory  of  the  Ley der»  Jar,  .  .412 
Why  the  jar  accumulates  the  fluid,  .  412 
Effect  of  thickness  in  the  jar,  .  .  412 
Why  the  jar  must  be  uninsulated,  .  413 
Phenomena  of  the  jar  explained  on 
both  theories,  .  .  .  .413 


ELECTRICAL   LIGHT. 

Light  not  a  constant   attendant  of 

electricity, 414 

Rules  for  increasing  the  vividness  of 

the  light  from  the  machine,  .         .  414 
Relation  of  the  length,  color  and  form 

of  the  spark  to  different  conductors,  415 
Illuminated  chain,     ....  415 
Passage  of  the  spark  through  rare- 
fied air, 415 

Do.  through  the  Torricellian  vacuum,  416 
Phenomena  of  the  spark  when  pass- 
ed through  the  vapor  of  ether,  al- 
cohol, &c 416 

Experiments  of  Davy  on  the  electric 

spark  in  vacuo,      ....  416 
Appearance  of  the  spark  in  conden- 
sed air, 417 

Do.  in  ivory,  sugar,  fluor  spar,  &c.  .417 
Illuminated  words,  ....  418 
Source  of  the  electric  light,  .  .  418 
Its  identity  with  solar  light,  .  .  419 


ELECTRIC    BATTERY. 


Battery  described,     . 
Object  of  the  battery, 
Great  battery  at  Haarlem, 
Effects  of  this  battery,      . 


.  419 
.-419 
.  420 
.  420 


MECHANICAL   EFFECTS   OF   ELECTRICITY. 

Origin  of  the  report, ....  421 
Rending  of  imperfect  conductors,  .  421 
Effect  on  a  quire  of  paper,  .  .  421 
Expansion  of 'fluids,  .  .  .  421 
Do.  of  confined  air,  .  .  .  .421 

CHEMICAL    EFFECTS   OF   ELECTRICITY. 

Enumeration  of  these  effects,  .  .  422 
Combustibles  inflamed,  .  .  .  422 
Oxides  revived,  .  .  .  .422 

MOTIONS    OF   THE    ELECTRIC    FLUID. 

Velocity  instantaneous,  .  .  .  422 
Preference  of  the  best  conductors,  .  423 
Preference  of  a  shorter  route,  .  .  423 
Influence  of  points,  ....  423 

EFFECTS    OF   ELECTRIC.  TV   ON   ANIMALS. 

When  is  the  shock  received  ?  .  .423 
Method  of  administering  the  shock,  .  424 
Phenomena  produced  by  shocks  of 

various  degrees  of  intensity, .         .  424 
To  communicate  the  shock  to  a  num- 
ber of  persons  at  once,  .         .         .  424 
Effects   of  the   shock  when  passed 
through  the  diaphragm,         .         .  424 


12 


ANALYSIS. 


Effects  of  the  charge  when  received 

on  an  insulating  stool,  .  .  .  425 
Lane's  Discharging  Electrometer  de- 

scribed 425 

To  communicate  the  shock  to  any 

particular  part  of  the  system,  .  425 
Animals  killed  by  electricity,  .  .  426 

Medical  electricity 427 

Medicated  Tubes,  .  .  .  .427 
Medicinal  properties  of  electricity,  .  427 

CAUSE    OF   ELECTRICAL    PHENOMENA. 

Use  of  the  term  electric  fluid,  .  .  428 
Reasons  for  believing  in  the  existence 

of  such  a  fluid,  ....  428 
Points  in  which  the  two  hypotheses 

concur,  .  .  •  I  •  •  429 
Arguments  in  favor  of  only  one 

fluid, 430 

Do.  in  favor  of  two  fluids,  .  .431 

ATMOSPHERICAL   ELECTRICITY. 

Atmosphere  always  electrified,  .  433 
Electricity  of  the  upper  regions,  how 

examined, 433 

Electrical  Kite  described,  .  .  433 
Experiments  of  Cavallo  with  it,  .  433 
Apparatus  of  Romas,  .  .  .  434 
Analogies  between  electricity  and 

lightning, 43.r> 

Identity  first  proved  in  France,         .  43G 
Experiments  of  Franklin,          .         .  436 
Source  of  atmospherical  electricity,  .  436 
Evaporation  and  condensation,  com- 
bustion, friction  of  opposite  winds,  437 


Leading    Facts. — Thunder     clouds 

highly  charged,  ....  437 
Hot  weather  succeeding  rainy  days,  438 
Wet  weather  succeeding  dry  and  hot 

days 438 

Electricity  of  clear  and  steady  wea- 
ther,  438 

Combination  of  the  elements  in  thun- 
der storms, 438 

Season  of  the  year,  .  .  .  .438 
Quarter  from  which  they  come,  .  438 
Thunder  and  lightning  of  volcanoes,  438 
THunder  storms  on  the  sea  and  on 

mountains, 438 

Peculiar    succession    of    lightning, 

thunder  and  rain, ....  438 
Explanation  of  thunder  storms,  .  438 
Rapid  formation  and  condensation  of 

vapor, 439 

Variability  of  the  winds,  .  .  .439 
Returning  stroke  explained,  .  .  439 
Theory  applied  to  the  leading  facts,  440 


LIGHTNING    ROI«. 

Construction  of  rods  pointed  out,      .  442 

PRECAUTIONS   FOR   SAFETY. 

Less  damage  in  a  city  than  in  inf- 
lated situations,     ....  44!t 
Why  ships  are  not  more  frequently 

struck, 443 

Why  barns  are  often  struck,     .         .  443 
Silk  dresses,  how  far  they  afford  pro- 
tection,   444 

Case  of  a  feather  bed,  .  .  .444 
Tall  trees  near  a  dwelling  house,  .  444 
Chimneys  to  be  especially  protected,  444 
Danger  from  being  near  small  collec- 
tions of  water,  ....  444 
Situations  peculiarly  safe,  .  .  445 

ANIMAL   ELECTRICITY. 

Torpedo — notices  of  the  ancients,  .  445 
Experiments  of  Davy,  .  .  .  446 
Torpedo  described,  ....  446 
His  electrical  powers  explained,  .  446 
Gymnotus — where  found, .  .  .  447 

Do.  described 447 

Affords  a  spark,  .  .  .  .447 
Attack  upon  horses, ....  447 
Silurus  Electricus,  .  .  .  .447 
Electricity  of  furred  animals,  .  .  447 
Spontaneous  appearances  of  electri- 

cal  light, 447 

Disposition  to  ascribe  too  many 

things  to  the  agency  of  electricity,  448 
Its  relation  to  chemical  affinity,  .  448 

MAGNETISM. 

Definition  of  magnetism,  .         .         .  450 

Magnets  defined 450 

Loadstone  described,  .  .  .  450 
Its  attractive  powers  long  known,  .  450 
Its  directive  powers,  when  discovered,  450 
Definitions — poles — axis,  .  .  .  451 
General  properties  enumerated,  .  451 

MAGNETIC    ATTRACTION. 

Attraction  of  the  magnet  for  iron,  .  452 
The  action  reciprocal,  .  .  .  452 
Other  metals  which  are  attracted,  .  452 
General  law  of  attraction  and  repul- 
sion between  the  poles, .  .  .  452 
How  iron  is  rendered  magnetic  by 

induction,  .  «.  .  .  .  453 
Opposite  polarity  induced  upon  the 

two  ends  of  a  bar,  .  .  .  453 
Position  of  two  magnetic  bars  for  a 

maximum  effect 454' 

Power  of  a  magnet  increased  by  action,  454 


13 


Action  of  a  strong  magnet  on  the 
poles  of  a  weak  one,  .  .  .  454 

Effect  when  the  north  pole  of  a  mag. 
net  is  placed  on  the  center  of  an 
iron  bar, 454 

Do.  when  placed  in  the  center  of  a 
disk  terminating  at  the  periphery 
in  stars, 454 

Case  of  a  bar  placed  between  the  dis- 
similar poles  of  two  magnets,  .  454 

Iron  when  affected  by  the  magnet 
becomes  itself  a  magnet,  .  .  455 

Comparative  power  of  soft  iron  and 
hardened  steel 455 

Several  pairs  of  poles  in  a  long  bar 
affected  by  a  strong  magnet,  .  455 

Causes  which  accelerate  the  progress 
of  magnetizing,  ....  455 

Do.  which  impair  or  destroy  magnet- 
ism, .  .  .  .  .  .  456 

A  magnetic  bar  divided  into  parts, 
each  part  has  polarity,  .  .  .  456 

Law  of  magnetic  attraction  and  re- 
pulsion in  relation  to  distance,  .  457 

Magnetic  power  resides  on  the  sur- 
face,   457 

DIRECTIVE    PROPERTIES   OF   THE    MAGNET. 

Case  of  a  needle  placed  near  the  pole 

of  a  magnetic  bar,          .         .         .  457 
Do.  of  a  needle  placed  at  right  angles 
to  the  bar,  directed  towards   the 
center  of  the  bar,  .         .         .         .457 

Curves     exhibited    by    iron     filings 
around  a  magnet, ....  458 

Nature  of  magnetic  curves,        .         .  458 
Magnetism  detected  in  various  bodies,  459 
Declination  or  variation  of  the  needle,  459 
Magnetic  meridian  and  equator,         .  460 
Line  of  no  variation,          .         .         .  460 
Discovery  of  the  variation  of  the  nee- 
dle,        .         .        .        .        .         .460 

Changes  in  the  variation  of  the  needle,  460 
Line  of  no  variation  traced,  .  .  461 
Amount  of  variation  in  different  pla- 

ces, 462 

Diurnal  variation,  ....  462 
Dip  of  the  needle,  ....  462 
Dipping  needle  described,  ,  .  .  463 
Position  of  the  magnetic  meridian,  .  463 
Magnetic  intensity  defined,  .  .  463 
Do.  how  determined,  .  .  .  464 
The  earth  itself  a  magnet,  .  .  464 
Cause  of  the  magnetism  of  the  earth,  464 
Magnetism  of  the  solar  rays,  .  .  465 
Analogy  between  Magnetism  and 

Electricity, 465 

Theory  of  magnetism,       .         .         .  466 
Magnetic  powers  of  iron  explained,  .  466 
Why   sofl.   iron   receives   and    loses 
magnetism  so  readily,  .        .        .  467 


METHODS   OF    MAKING   ARTIFICIAL  MAGNETS. 

Principles  enumerated,  .  .  .  467 
Magnetizing  by  contact,  .  .  .  468 

Armatures, 468 

Horse-shoe  magnet  described,  .  .  469 
Do.  how  made,  .  .  .  .469 
Rater's  rule  for  making  magnets,  .  469 
Best  material  for  compass  needles,  .  470 
Best  form  of  the  needle,  .  .  .470 
Best  mode  of  tempering,  .  .  .470 
Directive  forces  how  related  to  the 

length, 470 

Do.  the  mass, 470 

Power  of  iron  depends  on  the  surface,  470 
Reasons  for  the  various  precautions 

used  in  magnetizing,  .  .  .  470 
To  prevent  magnets  from  losing  their 

powers, 471 

Compass — form  of  the  needle,  .  .  472 
Surveyor's  compass  described, .  .  473 
Properties  of  thin  slender  needles,  .  473 
Mariner's  compass  described,  .  .  473 

LOCAL   ATTRACTION    OF  V 


Cause  of  the  irregularities  of  a  nee- 
dle on  board  ship, ....  474 
Barlow's  experiments,       .        .         .  475 
Experiments  on  ships  to  determine 

the  amount  of  local  attraction,  .  476 
Correcting  Plate  described,  .  .  476 
Chronometers  affected  by  magnetism,  477 
Magnetic  charts,  ....  478 
Aurora  Borealis — its  connexion  with 
magnetism,  .  .  .  .  •»  ? 


Optics  defined,         .         .         .         .480 
Objects  enumerated,          .         .        .  480 
Luminous  bodies  of  two  kinds,          .  480 
Definitions — ray,  beam,  pencil,  medi- 
um, transparent,  semi-transparent, 
translucent,  opake,         .        .         .  480 
Rectilinear  direction  of  light,    .        .  481 
Velocity  of  light,       .  .481 

Methods  of  determining  it,         .         .  482 
Intensity  of  light  at  different  distan- 
ces from  the  radiant,     .         .         .  482 
Rate  at  which  a  transparent  medi- 
um would  diminish  the  intensity  of 

light, .483 

Figure  of  the  shadow  of  globes,       .  483 

REFLEXION  OF   LIGHT. 

Definition  of  reflexion,     .         .         .  484 
Law  of  reflexion,      .        .        .         .485 
Mode  of  conducting  optical  experi- 
ments,   485 

Reflexion  from  plane  mirrors,  .        .  486 


14 


ANALYSIS. 


Parallel,  Diverging,  and  Converging 
rays, 486 

Concave  mirrors,  how  they  reflect 
parallel,  diverging,  and  converging 
rays,  .  ...  487 

Convex  mirrors — reflexion  of  the  sev- 
eral sorts  of  rays,  .  .  .  489 

IMAGES  FORMED  BY  MIRRORS. 

Image  formed  by  plane  mirrors,  .  490 
Place  of  the  imaginary  radiant,  .  490 
Distance  of  the  image  behind  the 

mirror, 491 

Angular   deviation  of  an  object  re- 
flected from   two  mirrors  inclined 
to  each  other,         .         .         .         .491 
Principle  of  Hadley's  Quadrant,        .  492 
Ratio  of  the  length  or  breadth  of  the 
object  to  that  of  the  part  of  the 
mirror  upon  which  the  image  ap- 
pears,     492 

Size  of  the  mirror  required  to  see  the 

entire  person,         ....  493 
Reflexion  from  parallel  mirrors,         .  493 
Do.  from  inclined         do.          .         .  494 
Proportion  of  rays  reflected  perpen- 
dicularly from  water,     .         .         .496 
Inverted  image,  when  produced,       .  496 
Images  formed  by  concave  mirrors,    497 
When  the  object  is  between  the  mir- 
ror and  the  focus  of  parallel  rays,    497 
When  the  object  is  in  the  focus,         .  497 
When  the  object  is  between  the  fo- 
cus and  the  center,        .        .         .  497 
When  tke  object  is  beyond  the  cen- 
ter,         497 

When  the  object  is  in  the  center,      .  497 
Images  formed  by  convex  mirrors,     .  498 
In  spherical   mirrors,   ratio    of    the 
diameter  of    the  object   and   the 

image, 499 

Various    appearances   presented  on 

looking  into  a  concave  mirror,  .  499 
Caustics  by  reflexion  described,  .  499 
Use  of  concave  mirrors  by  showmen,  500 
Use  for  light-house  reflectors,  .  .501 
Large  burning  mirrors,  .  .  .  501 
Experiments  of  Archimedes,  .  501 

REFRACTION  OF    LIGHT. 

Refraction  defined,  .  .  .  .502 
Angles  of  incidence  and  refraction,  502 
Ratio  of  their  sines,  .  .  .  502 
Refraction  of  water,  sulphur,  and 

diamond  compared,  .  .  .  503 
Index  of  refraction  defined,  .  .  503 
When  a  ray  of  light  cannot  pass  out 

of  a  denser  into  a  rarer  medium,  504 
Angle  of  total  reflexion,  .  .  .504 
Refractive  powers  of  various  bodies,  504 


Highest  refracting  be  lies,          .         .505 
Lowest,  do.      ...  .  505 

Prism  described,       ....  506 
Index  of  refraction  compared  with 

the  angle  of  deviation,  .         .  507 

To  find  the  index  of  refraction  of  a 

body, 507 

Refraction  through  a  medium  bound- 
ed by  plane  and  parallel  surfaces,     507 
Refraction  from  a  rarer  into  a  denser 
medium,  or  from  a  denser  into  a 

rarer, 508 

Lenses  enumerated,          .         .         .  508 

Axis  of  a  lens, 509 

The  several  cases  of  rays  passing 
out  of  one  medium  into  another, 
through  spherical  surfaces,    .         .  509 
General  office  of  a  convex  lens,         .  512 
Do.  of  a  concave  lens,      .         .         .  512 
Manner  of  forming  images  by  a  lens,  513 
Ratio  of  the  diameter  of  the  object 

to  that  of  the  image,  .  .  .  514 
Spherical  aberration  defined,  .  .  514 
Dimensions  of  the  image  not  affect- 
ed by  changing  the  area  of  the  lens,  515 
How  the  brightness  is  affected,  .  515 
Amount  of  spherical  aberration  in 

several  kinds  of  lenses,          .         .  515 
Form  of  a  lens  which  has  no  spheri- 
cal aberration,       ....  516 
Practical    objections   to   the  use   of 
such  lenses,  .        .        .        .        .516 

DECOMPOSITION  OF   LIGHT. 

Composition  of  solar  light,         .         .517 

Prismatic  spectrum,  how  formed,     .  518 

Colors  of  the  spectrum.     .         .         .  518 

Separate  colors  magnified  by  a  con- 
cave lens, 518 

Individual  colors  incapable  of  de- 
composition, ....  519 

Reflected  light  and  the  light  of  com- 
bustibles decomposed,  .  .  .  519 

Fixed  proportion  of  all  the  rays  es- 
sential to  white  light,  .  .  520 

Colors  compounded  of  two  different 
colors, 520 

Whether  the  colors  of  the  spectrum 
are  seven  in  number,  .  .  .521 

Experiments  of  Brewster  on  absorp- 
tion, .... 

Fixed  lines  in  the  spectrum, 

NATURE    OF   LIGHT. 


522 
.  523 


Two  hypotheses,      .        .         .        .524 

Each  stated 525 

Objections    to  the    Undulatory  hy- 
pothesis  525 

Ditto  to  the  Emission  do.  .         .        .  526 
Which  late  discoveries  favor,    .        .  527 


15 


COLORS   IN   NATURAL   OBJECTS 

Page- 

Rainbow  described,  ....  527 

Ray  of  light  traced  through  a  drop 
of  water,  .  .  .  .  .  528 

Accumulation  of  rays  in  a  certain 
part  of  the  drop,  .  .  .  529 

Position  of  the  spectator  with  re- 
spect to  the  sun  and  the  bow,  .  530 

How  the  inferior  bow  is  produced,     .  530 

How  the  superior  bow  is  produced,    .  531 

Width  of  the  colored  bow,         .         .531 

Order  of  the  colors  in  each  bow,       .  532 

How  the  axis  of  vision  passes  with 
respect  to  the  bow,  .  .  .  532 

Height  of  the  bow  when  the  sun  is  on 
the  horizon, 532 

Circular  rainbows  seen  on  high  moun- 
tains explained,  ....  532 

Colors  of  bodies — cause  of  various 
colors,  ......  532 

Newton's  experimerts  on  colored 
rings,  .  .  .533 

Relation  of  particular  colors  to  cer- 
tain degrees  of  thickness  of  the 
medium,  .....  534 

Explanation  of  the  phrases,  fits  of 
easy  reflexion  and  transmission,  .  536 

Newton's  experiments  on  soap  bub. 
bles, 536 

Summary  of  the  Newtonian  doctrine 
of  colors, 537 

Cause  of  opacity,      .        .  .  537 

INFLEXION  OR    DIFFRACTION  OF   LIGHT, 

Inflexion  donned,      .         .         .         .538 

Mode  of  performing  the  experiment,  538 

Phenomena  of  colored  fringes,           .  538 
Appearance  of  a  pencil  of  light  pass- 
ing  through  a  small  aperture,  when 

viewed  by  a  magnifier,          .         .  539 

Cause  of  the  inflexion  of  light,          .  539 

Doctrine  of  interferences,          .         .  540 

DOUBLE    REFRACTION   AND  POLARIZATION. 

Double  refraction  defined,          .        .  541 

Bodies  which  exhibit  it,     .         .         .  541 
Phenomena  produced  by  the  Iceland 

crystal, 542 

Ordinary  and  extraordinary  ray,         .  543 
Optic  axis,         .         .         ....,'            .543 

Principal  section,      ....  543 

Positive  and  negative  axis,        .         .  543 
Axis  of  double  refraction  a  fixed  di- 
rection,            543 

Crystals  with  two  axes  of  double  re- 
fraction,    544 

Bodies  rendered  capable  of  dou- 
ble refraction  by  heat,  pressure, 

&c.,              544 


POLARIZATION  OF  LIGHT. 

Fife. 

Polarization  defined,  .  .  .  545 
Kinds  of  light  that  possess  no  po.ari. 

zation, 545 

Property  of  light  that  has  undergone 

double  refraction,  ....  545 
Changes  produced  by  polarization 

explained, 546 

Produced  by  refraction  and  reflexion,  546 

Polarizing  angles  for  glass,  .  .  547 

Relation  of  color  to  polarized  light,  548 


Circular  image   of  the  sun  formed 
by  admitting  a   sunbeam  into    a 

darkroom, 549 

Whether  the  shape  of  the  aperture 

is  material,    .....  549 
Images  of  the  sun  in  eclipse,  .         .  550 
Camera  Obscura,  how  formed,          .  550 
Circumstances  essential  to  the  bright- 
ness and  distinctness  of  the  pic- 
ture,       550 

Effect  of  a  lens  in  perfecting  the  pic- 
ture,   551 

Scioptic  Ball  described,  .  .  .  551 
The  Eye — its  analogy  to  the  camera 

obscura, 552 

Eye  described,  ....  552 
Aqueous  humor,  ....  552 
Cornea — its  figure,  ....  553 
Figure  of  the  aqueous  humor,  .  .  553 
Iris — pupil — its  uses  and  varieties,  .  553 
Crystalline  lens — its  figure  and  of- 
fice,   553 

Unequal  density  of  the  crystalline,  554 
Vitreous  humor,  ....  554 
Retina — pigmentum  nigrum,  .  .  554 

Sclerotica, 554 

Use  of  the  cornea  in  enlarging  the 

field  of  view,         .         .         .         .555 
Great  range  of  vision,      .         .         .  555 
How    the    image    remains   distinct 
while  objects  are  at  different  dis- 
tances,         .....  555 
Adaptation  of  the  eyes  of  different 
animals  to  their  peculiar  circum- 
stances,          556 

Why  an   inverted  image   represents 

objects  erect,  ....  556 
Cause  of  cata/act  and  its  remedy,  .  557 
Case  of  short-sighted  persons,  .  .  557 
Thaumatrope  and  Phantasmascope,  558 
Why  objects  do  not  appear  double 

when  seen  with  two  eyes,  .  .  559 
How  objects  are  made  to  appear 

double 559 

Minute  objects  best  seen  by  a  side  view,  560 
Distances  and   Magnitudes,  how  es- 
timated,        ...  .560 


16 


MICROSCOPES. 

Microscopes  defined,  .  .  .  562 
Simplest  form,  .  .  .  .562 

Limit  of  distinct  vision,  .  .  .  562 
Distinctness — how  produced,  .  .  562 
Increased  magnitude,  do.  .  .  563 
Mode  of  action  of  a  convex  lens,  .  563 
Magnifying  power — how  determined,  563 

Field  of  view, 564 

Aperture,  .         .        .        .        .564 

Diamond  and  Sapphire  Microscopes,  564 
Advantages  of  the  diamond  over 

glass, .564 

Why  diamond  lenses  are  achromatic,  565 
Magnifying  powers  of  Pritchard's 

diamond  microscopes,            .         .  565 
Fluid  microscopes,  how  formed,         .  566 
Substances  used  for  this  purpose,     .  566 
Perspective  Glass  described,     .         .  566 
Use  of  microscopes  to  form  magni- 
fied unages,  as  in  the  Magic  Lan- 
tern and  Solar  Microscope,    .         .  567 
Principles  recapitulated,    .         .         .  568 
Magnifying    power,    how  determin- 
ed,          569 

Magic  Lantern  described,  .  .  569 
Objects  represented,  .  .  .570 
Solar  microscope  described,  .  570 

Performances  of  this  instrument,  .  571 
Compound*  microscope,  its  construe. 

tion  and  principle,           .         .         .  572 
Magnifying  power,  how  estimated,  .  572 
Why  high  magnifiers  cannot  be  em- 
ployed,   573 

Large  field  of  view,  .         .         .         .573 
Lucernal  microscope  described,         .  573 
Reflecting  Microscopes,  how  formed,  574 
Rules  for  making  microscopic  obser- 
vations,           574 

Portable  Camera  Obscura  described,  575 
Camera  Lucida  described,  .  '  .  575 


TELESCOPES. 

Telescopes  defined,  .  .  .  57*7 
Leading  principle  enunciated,  .  .  577 
Variety  of  telescopes,  .  .  .  578 
Astronomical  Telescope  described,  .  578 
Its  analogy  to  the  compound  micro- 
scope,   578 

Magnifying  power,  how  determined,  579 
Image  inverted,  how  inade  erect,  .  579 
Difficulties  to  be  overcome,  .  .  580 
Spherical  Aberration,  its  cause  and 

amount, 580 

How  overcome 580 

Advantage  of  large  apertures,  .         .  580 
How  spherical  aberration  is  prevent- 
ed in  the  eye-glass,        .         .        .  581 
Do.  in  the  object-glass,      .         .         .581 
Effect  of  making  it  in  the  form  of  a 

paraboloid, 582 

Chromatic  Aberration,  its  cause,      .  58.2 

Dispersion  defined 583 

Dispersive  powers  of  bodies,  .  .  584 
How  the  object-glass  of  a  telescope 

is  rendered  achromatic,  .  .  584 
Structure  of  an  achromatic  lens,  .  585 
Perfection  of  the  Achromatic  telescope,  585 
Use  of  a  large  aperture,  .  .  .  585 
Want  of  light  and  field,  how  remedied,  586 
Imperfections  of  glass,  .  .  .  586 
Difficulty  of  obtaining  perfect  glass,  587 
Dorpat  Telescope,  .  .  .  .587 
Terrestrial  or  Day  telescopes  de- 
scribed, ^ 588 

Use  of  the  various  eye-glasses,  .  588 
Galileo's  Telescope  described,  .  .  589 
Its  advantages  and  disadvantages,  .  590 
Reflecting  Telescope  described,  .  591 
How  spherical  and  chromatic  aber- 
ration are  prevented,  .  .  .591 
Herschel's  great  telescope  described,  592 
Lord  Oxmanton's  do  .  .  592 


NATURAL  PHILOSOPHY. 


PART  . MATHEMATICAL  ELEMENTS  OP  MECHANICS. 

[ON   THE   BASIS   OF  BRIDGE'S   MEGHAN    --S.] 


PRELIMINARY    DEFINITIONS. 

ART.  1.  NATURAL  PHILOSOPHY  is  the  science  which  treats  of  the 
Laws  of  the  Material  world. 

This  is  the  primitive  signification  of  the  term  Natural  Philoso- 
phy ;  but  the  vast  extension  given  to  inquiries  into  the  laws  of 
nature,  rendered  a  division  of  them  necessary.  Hence,  those 
laws  of  nature  which  relate  to  masses  of  matter  were  retained 
by  Natural  Philosophy,  (which  has  been  farther  divided  into  Me- 
chanical Philosophy  and  Astronomy,)  while  those  which  relate  to 
particles  of  matter,  and  to  the  changes  of  constitution  produced 
by  their  action  on  each  other,  were  assigned  to  Chemistry. 

The  term  law,  as  here  used,  signifies  the  mode  in  which  the  powers 
of  nature  act.  Laws  are  general  truths,  comprehending  a  great 
number  of  subordinate  truths. 

Natural  or  Mechanical  Philosophy  is  divided  into  Mechanics, 
Hydrostatics,  Pneumatics,  Electricity,  Magnetism,  and  Optics. 

MECHANICS  is  that  branch  of  Natural  Philosophy,  which  treats  of 
the  Equilibrium  and  Motion  of  bodies.  As  the  changes  which  occur 
between  masses  of  matter,  involve  the  idea  of  motion,  hence,  the 
causes  which  produce  motion,  or  which  prevent  it,  and  the  man- 
ner in  which  it  takes  place,  (its  laws,)  constitute  the  great  object 
of  inquiry  in  mechanical  philosophy. 

Body  is  any  collection  of  matter  existing  in  a  separate  form. 

Force  is  any  cause  which  moves  or  tends  to  move  a  body,  or 
which  changes  or  tends  to  change  its  motion.  Every  force  pro- 
duces actual  motion  if  it  is  not  counteracted  by  contrary  forces ; 
but  if  it  remains  counteracted,  the  motion  which  it  tends  to  pro- 
duce is  called  Virtual. 

That  part  of  Mechanics  which  relates  to  the  action  of  forces 
producing  equilibrium  or  rest,  in  bodies,  is  called  Statics;  that 
which  relates  to  the  action  of  forces  producing  motion,  is  called 
Dynamics.* 

*  In  the  following  treatise,  it  is  found  convenient  to  disregard  this  distinction. 

3 


18  NATURAL   PHILOSOPHY. 

The  science  of  Mechanics  comprehends  those  laws  of  equilib- 
rium and  motion  only,  which  are  common  to  all  bodies  in  the  uni- 
verse, and  to  bodies  in  every  form,  whether  solid,  fluid,  or  aeri- 
form ;  but  the  laws  of  equilibrium  and  motion  undergo  certain 
additional  modifications  in  consequence  of  the  peculiar  properties 
of  fluids.  Hence  that  branch  of  Mechanics  which  treats  of  the 
peculiar  mechanical  properties  of  fluids  in  the  form  of  water,  is 
called  Hydrostatics  ;  and  that  which  treats  of  the  peculiar  me- 
chanical properties  of  fluids  in  the  form  of  air,  is  called  Pneumatics. 

2.  The  two  essential  properties  of  matter,  both  of  which  are  in- 
separable from  it,  are  extension  and  impenetrability.  Extension, 
in  the  three  dimensions  of  length,  breadth,  and  thickness,  belongs 
to  matter  under  all  circumstances ;  and  impenetrability,  or  the 
property  of  excluding  all  other  matter  from  the  space  which  it  occu- 
pies, appertains  alike  to  the  largest  body  and  the  smallest  particle. 
The  word  particle  is  much  used  in  writings  on  physical  subjects. 
In  Natural  Philosophy  we  mean  by  particles,  the  smallest  parts 
into  which  a  body  may  be  supposed  to  be  divided  by  mechanical 
means,  without  any  reference  to  the  different  elements  of  which 
such  particles  may  be  composed.  Inquiries  respecting  these  be- 
long to  Chemistry. 

The  quantity  of  matter  which  a  body  contains,  is  called  its 
]\fass ;  the  space  it  occupies,  its  Volume  ;  its  relative  quantity  of 
matter  under  a  given  volume,  its  Density.  All  bodies  have  empty 
spaces  denominated  Pores.  In  solids,  we  may  often  see  the  pores 
with  the  naked  eye,  and  almost  always  by  the  microscope ;  in 
fluids,  their  existence  can  be  proved  by  experiment.  The  ratio 
of  the  space  occupied  by  the  pores  of  a  body  to  that  occupied  by 
the  solid  matter,  is  not  known ;  but  there  are  reasons  for  believing 
that  even  in  the  densest  bodies,  the  amount  of  solid  matter  is 
small  compared  with  the  empty  spaces.*  Hence  it  is  inferred 
that  the  particles  of  matter  touch  each  other  only  in  a  few 
points.f 

Although  extension  and  impenetrability  are  said  to  be  the  es- 
sential properties  of  matter,  because  they  are  inseparable  from 
its  very  existence,  yet  there  are  also  several  other  properties 
which  are  known  by  experience  to  belong  to  all  matter,  as  gravity, 
inertia,  and  divisibility ;  and  others  still  which  belong  not  to  mat- 
ter universally,  but  only  to  certain  classes  of  bodies,  as  elasticity, 
or  the  power  a  body  has  of  recovering  itself  when  compressed ; 
malleability,  or  the  power  of  being  extended  into  leaves  or  plates ; 
and  ductility,  or  the  power  of  being  extended  in  length,  as  when 
drawn  into  wire. 

In  Geometry,  we  conceive  figures  to  possess  extension  only  with- 
out solidity ;  or  to  occupy  space  without  excluding  other  figures 

*  See  Newton's  Optics,  Lib.  II,  iii,  Pr.  8.  t  Playfair. 


MECHANICS.  19 

from  it ;  but  in  Mechanics,  we  take  objects  as  they  occur  in 
nature,  viz.  not  only  extended,  but  impenetrable* 

CHAPTER  I. 

OF  MOTION  AND  THE  LAWS  OF  MOTION. 

3.  ATTRACTION  is  the  tendency  which  one  portion  of  matter  has 
toward  another,  and  exists  both  between  particles  and  between 
masses  of  matter.     Aggregation  is  the  union  of  particles  of  the 
same  kind  in  one  body ;  as  of  the  particles  of  lead  in  a  musket 
ball.     Affinity  is  the  union  of  particles  of  different  kinds  in  one 
body ;  as  of  particles  of  copper  and  of  zinc  in  brass.     Cohesion  is 
the  union  of  compound  particles  in  one  body ;  thus  a  particle  of 
copper  and  a  particle  of  zinc  are  united  to  form  a  particle  of  brass 
by  affinity,  and  the  particles  of  brass  are  united  by  cohesion.     In 
mechanical  philosophy,  however,  the  term  cohesion  is  usually 
employed  to  denote  the  union  of  particles  of  all  sorts,  whether 
simple  or  compound,  leaving  to  Chemistry  all  inquiries  respecting 
the  composition  of  bodies.     Gravity  is  that  property  by  which  aU 
terrestrial  bodies  tend  toward  the  center  of  the  earth.     It  is  in 
this  sense  that  gravity  is  understood  as  a  force  in  Mechanics.   But 
in  order  to  give  the  learner  correct  views  of  this  important  sub- 
ject, we  subjoin  a  few  other  particulars  respecting  it. 

4.  Gravity  is  a  property  of  matter,  universally ;  and  the  force  of 
gravity  in  any  body  is  proportioned  to  its  quantity  of  matter. 

Since  every  particle  of  matter  is  endowed  with  this  property, 
it  follows  that  the  force  of  gravity  is  proportioned  to  the  mass  or 
quantity  of  matter,  f  We  do  not  say  what  gravity  is,  but  what 
it  does, — namely,  that  it  is  something  which  gives  to  every  par- 
ticle of  matter  a  tendency  toward  every  other  particle.  This 
influence  is  conveyed  from  one  body  to  another  without  any  per- 
ceptible interval  of  time.  J  Gravity  extends  to  all  known  bodies 
in  the  universe,  from  the  smallest  to  the  greatest ;  but  the  con- 
sideration of  the  subject,  in  this,  extent,  belongs  to  Astronomy. 
We  at  present  contemplate  gravity  only  as  it  affects  terrestrial 
bodies.  By  it  all  bodies  are  drawn  toward  the  center  of  the 
earth,  not  because  there  is  any  peculiar  property  or  power  in  the 

*  Whewell's  Mechanics,  p.  2. 

t  A  decisive  proof  that  the  force  of  gravity  is  always  proportioned  to  the  quantity 
of  matter,  is  furnished  by  the  pendulum,  its  vibrations  (which  depend  on  gravity  and 
measure  its  force,  as  will  be  seen  hereafter)  being  always  performed  in  the  same  time, 
of  what  material  soever  it  is  made. — Francaeur,  Mech.  p.  63. 

t  If  the  action  of  gravitation  is  not  instantaneous,  it  moves  more  than  fifty  millions 
of  times  faster  than  light. — Fourier,  Eulogy  on  Laplace. 


20  NATURAL   PHILOSOPHY. 

center,  but  because,  the  earth  being  a  sphere,  the  aggregate  effect 
of  the  attractions  exerted  by  all  its  parts  upon  any  body  exterior 
to  it,  is  such  as  to  direct  the  body  toward  the  center ;  as  will  be 
more  fully  explained  hereafter. 

5.  This  property  discovers  itself,  not  orJy  in  the  motion  of  fall- 
ing bodies,  but  in  the  pressure  exerted  by  one  portion  of  matter 
upon  another  which  sustains  it ;  and  bodies  descending  freely  un- 
der its  influence,  whatever  be  their  figure,  dimensions,  or  texture, 
are  all  equally  accelerated  in  right  lines  perpendicular  to  the  plane 
of  the  horizon.  The  apparent  inequality  of  the  action  of  gravity 
upon  different  species  of  matter  near  the  surface  of  the  earth, 
arises  entirely  from  the  resistance  which  they  meet  with  in  their 
passage  through  the  air.  When  this  resistance  is  removed,  (as 
in  the  exhausted  receiver  of  an  air-pump,)  no  such  inequality  is 
perceived ;  bodies  of  all  kinds  there  descend  with  equal  velocities ; 
and  a  guinea,  a  feather,  and  the  smallest  particle  of  matter,  if  let 
fall  together,  are  observed  to  reach  the  bottom  of  the  receiver 
exactly  at  the  same  instant. 

G.  The  weight  of  a  body  is  the  force  it  exerts  in  consequence 
of  its  gravity,  and  is  measured  by  its  mechanical  effects,  such  as 
bending  a  spring,  or  turning  a  balance.  The  force  thus  exerted 
by  a  given  mass  of  matter,  (as  a  cubic  foot  of  water,)  being  ta- 
ken as  the  standard,  called  1000,  and  accurately  counterpoised  in 
a  balance  by  some  substance  easily  susceptible  of  division,  (as  a 
mass  of  lead  for  example,)  multiples  or  aliquot  parts  of  this  stand- 
ard weight,  afford  the  means  of  estimating  the  weights  of  all 
other  bodies.  We  weigh  a  body  by  ascertaining  the  force  required 
to  hold  it  back,  or  to  keep  it  from  descending.  Hence,  weights  are 
nothing  more  than  measures  of  the  force  of  gravity  in  different 
bodies ;  but  since  the  force  of  gravity  is  proportioned  to  the 
quantity  of  matter,  (Art.  4,)  weights  are  also  measures  of  the 
comparative  quantities  of  matter  in  different  bodies. 

7.  Gravity  at  different  distances  from  the  Earth,  varies  inversely 
as  the  square  of  the  distance  from  its  center. 

The  total  amount  of  attraction  exerted  by  the  earth  upon  bod- 
ies exterior  to  it,  is  the  same  as  though  that  force  were  all  con- 
centrated in  the  center.  (Art.  4.)  But  a  force  or  influence 
which  proceeds  in  right  lines  from  a  point  in  every  direction, 
is  diminished  as  the  square  of  the  distance  is  increased.  For, 
let  S  be  the  center  of  the  earth;  and  since  the  force  of 
gravity  acts  in  right  lines  directed  towards  that  center,  what- 
ever be  the  nature  of  gravity,  its  influence  at  the  distance  SA, 
will  be  equally  diffused  over  the  surface  ABCD  ;  and  at  the  dis- 
tance SE,  it  will  be  equally  diffused  over  the  surface  EFGH. 
Therefore  its  intensity  or  force  will  be  as  much  less  at  the  point 


21 


E  than  at  A,  as  EFGH  is  greater  than  ABCD  ;  that  is,  the  force 
of  gravity  at  A,  is  to  the  force  of  gravity  at  E,  as  EFGH  is  to 
ABCD.  But, 

EFGH  :  ABCD  :  :  EF2  :  AB2  :  :  SE2  :  SA2. 

/.Force  of  Gr.  at  A  :  Force  of  Gr.  at  E  :  :  SE2  :  SA2  ;  or  the 
forces  of  gravity  at  A  and  E  are  inversely  as  the  squares  of  the 
distances  from  the  center. 

8.  The  weight  of  a  body,  therefore,  will  vary  at  different 
heights  above  the  earth's  surface.  Thus,  at  twice  the  distance 
from  the  center,  or  at  the  height  of  about  4000  miles  above  the 
earth,  the  force  of  gravity  is  only  one  fourth  as  great  as  at  the 
surface,  and  a  given  body  would  weigh  only  one  fourth  as  much 
as  at  the  earth.  The  moon  being  60  times  as  far  from  the  earth's 
center,  as  the  distance  from  that  center  to  the  surface,  the  attrac- 
tion of  the  earth  upon  the  moon  is  3600  (=602)  times  less  than 
upon  bodies  near  the  earth.  But  the  heights  at  which  experi- 
ments are  commonly  made  upon  the  weights  of  bodies,  bear  so 
small  a  ratio  to  the  radius  of  the  earth,  that  this  variation  is  com- 
monly imperceptible.  At  the  height  of  half  a  mile,  the  diminu- 
tion does  not  amount  to  more  than  about  7oVo^h  part  of  the 
weight  at  the  surface.  For,  let  r=the  radius  of  the  earth=4000 
miles,  nearly  ;  and  let  x  be  the  height  of  the  body,  W  its  weight 
at  the  earth's  surface,  and  W  its  weight  at  the  height  x.  Then, 

W  :  W  :  :  (r+x)2  :  r2  :  :  r2+2j,*+*2  :  r2. 
W  :  W-W  :  : 


But  when  a;  is  a  small  fraction  of  r,*  #2  may  be  neglected,  and 
then,  W  :  W  :  :  r*+2rx  :  r2  :  :  r+2x  :  r, 

W  :  W-W  :  :  r+2x  :  2x  .'.  W-  W'=^-^(B) 

Let  x  be  half  a  mile  ;  then    •*     =  4  oVrth  part  of  the  whole 

weight  ;  or,  a  body  would  weigh  so  much  less  at  the  height  of 
half  a  mile  than  at  the  surface  of  the  earth.     But  if  the  height 

»  If,  for  example,  *=  JL  of  r,  or  J  a  mile,  then  **=  (  JL)*  = 
quantity  so  small  that  it  may  be  neglected. 


22  NATURAL    PHILOSOPHY. 


were  100  miles  above  the  earth,  then  tVV^iV  »  and  the  square 
of  this=T?V?  °f  the  radius  of  the  earth,  a  quantity  too  large 
to  be  neglected  ;  and  the  difference  of  weights  at  the  surface  and 
at  the  height  of  100  miles,  will  be  found  by  formula  (A.) 

What  loss  of  weight  would  a  body  sustain,  by  being  elevated  500 
miles  above  the  earth  ?  Ans.  -J-f  .  Were  ar2  neglected,  then  the  loss 
by  formula  (B)  would  be  j,  which  differs  from  \\  by  only  ?£j. 

9.  A  body  situated  within  a  hollow  sphere,  would  remain  at  rest  in 
any  part  of  the  void.* 

Let  CAD  represent  the  surface  of 
a  hollow  sphere,  and  P  any  point  in 
the  void.  Through  P  let  the  plane 
CD  pass,  dividing  the  sphere  into  any 
two  segments  CAD  and  CaD. 

Let  PBA  and  ~Pba  represent  two 
cones  meeting  in  a  very  small  angle 
at  P,  and  having  their  bases  in  the 
surface  of  the  sphere  ;  which  bases, 
being  indefinitely  small,  may  be  con- 
sidered as  plane  figures,  and  being  cir- 
cles, they  are  to  one  another  as  the 
squares  of  their  diameters  ;  that  is,  as  AB2  to  ab2.  But  when  BA 
and  ba  are  indefinitely  small,  the  lines  PB  and  PA  may  be  c*bnsid- 
ered  equal,  as  also  Pa  and  P6.  Therefore,  the  two  triangles  PAB 
and  Pa6  are  similar,  and  AP2  :  aP2  :  :  AB2  :  «&2.  But,  putting  Q, 
for  quantity  of  matter,  D  for  distance,  and  G  for  gravity, 
AB2  :  «&2  :  :  base  of  PAB  :  base  of  fab  :  :  Q  :  ?.f 

.-.  AP2  :  <zP2  :  :  Q  :  q  :  :  D2  :  d*l  /.  Q  aD2  (1). 
But  since  the  force  of  gravity  varies  directly  as  the  quantity  of 
matter,  and  inversely  as  the  square  of  the  distance, 

Q  D2 

G  oc  ^  .*.  (1)  G  oc  .pp,  or  G  is  a  constant  quantity. 

Hence,  the  point  P  (or  a  body  at  P)  is  equally  attracted  towards 
AB  and  ab,  and  the  same  will  be  the  case  with  all  the  correspond- 
ing portions  of  the  two  opposite  segments.  The  same  reasoning 
evidently  applies  to  all  the  concentric  surfaces  or  laminae  of 
which  the  shell  of  the  sphere  may  be  supposed  to  be  made  up  ; 
therefore  a  body  situated  anywhere  within  the  shell,  being  at- 
tracted by  equal  and  opposite  forces,  would  remain  at  rest.§ 

10.  The  force  of  gravity  below  the  earth's  surface  is,  at  different 
distances  from  the  center,  directly  as  those  distances. 

Were  a  body  placed  at  the  center  of  the  earth,  being  attracted 

The  solid  part  of  the  sphere  is  supposed  to  be  throughout  of  uniform  density. 
t  The  surface  of  the  sphere  is  here  considered  as  a  thin  lamina  of  matter 
J  Since  here  D  and  d  are  the  same  as  AP  and  oP. 
§  See  Newton's  Principia,  Book  I,  Pr.  70. 


MECHANICS.  23 

equally  in  all  directions,  it  would  evidently  remain  at  rest ;  and 
were  it  siluated  at  any  point  between  the  center  and  the  surface, 
the  force  of  gravity  toward  the  center  would  be  diminished  by 
the  loss  of  the  attraction  of  the  matter  exterior  to  it;  for  the  mat- 
ter exterior  to  it  would,  by  Article  9,  have  no  effect  upon  it. 
Thus,  if  a  body  were  to  fall'through  a  hole  bored  from  the  earth's 
surface  to  its  center,  the  gravity  of  it  would  constantly  diminish, 
until,  at  the  center,  it  would  become  nothing. 

Let  P  be  a  body  situated  within  the 
earth  at  any  distance  D  from  the  cen- 
ter. Then  it  appears  by  the  last  arti- 
cle, that  the  gravity  of  P  toward  the 
center  would  not  be  affected  at  all  by 
the  shell  exterior  to  PQR,  and  that  P 
would  gravitate  only  by  the  force  exert- 
ed by  the  sphere  PQR.  But  this  force 
G  is  directly  as  the  quantity  of  matter 
in  PQR,  and  inversely  as  the  square  of 
the  distance  from  the  center.  That  is, 

Q  D3 

G  ocj^.     But  Q  oc  D3*  /.  G  ocjp  ocD.    Therefore,  the  gravity  of 

P  varies  as  its  distances  from  the  center  of  the  sphere.f 

11.  The  IXERTIA  of  matter  is  its  resistance  to  a  change  of  state 
whether  of  rest  or  motion.     The  inertia  of  a  body  at  rest  is  fhe  re- 
sistance to  be  overcome  to  bring  it  to  a  given  velocity ;  or,  in 
common  language,  "  to  start  it ;"  and  the  inertia  of  a  body  in 
motion,  is  the  resistance  it  makes  to  being  stopped,  after  the 
moving  force  is  withdrawn.     Thus  the  inertia  of  a  steamboat, 
while  getting  under  weigh,  requires  a  great  expenditure  of  force 
to  bring  the  boat  to  its  final  velocity ;  but  its  inertia  carries  it 
still  forward  after  the  engine  is  stopped.     Since  every  particle  is 
endued  with  this  property,  the  inertia  of  a  body  is  proportioned  to 
its  quantity  of  matter,  and  of  course  (Art.  6)  to  its  weight.     Inertia, 
however,  is  a  more  sure  criterion  of  the  quantity  of  matter  in  a 
body  than  weight  is ;  for  inertia  is  always,  and  under  all  circum- 
stances, the  same :  whereas  weight,  being  merely  the  measure 
of  gravity,  is  diminished  as  gravity  is  diminished ;  so  that  it  is 
less  on  the  tops  of  high  mountains,  than  at  the  general  level  of 
the  earth. 

12.  In  observing  the  phenomena J  connected  with  the  actual 
motion  of  a  body,  we  consider  the  space  over  which  it  moves,  the 

*  The  quantity  of  matter  in  spheres  being  as  the  cubes  of  their  radii. 

t  Principia,  Book  I,  Pr.  73. 

t  The  word  phenomena  is  much  used  in  Natural  Philosophy.  It  is  thus  defined  : 
The  phrase  Natural  Phenomena,  in  its  widest  acceptation,  denotes  any  effects  in  the 
material  part  of  the  creation,  addressed  to  one  or  more  of  the  senses.  (Parkinson's 
Mechanics,  p.  1.) 


24  NATURAL   PHILOSOPHY. 

time  of  its  motion,  and  the  velocity.  A  body  is  said  to  move  with 
uniform  velocity  when  it  describes  equal  spaces  in  equal  times. 
When  the  spaces  described  by  it  in  e.qual  portions  of  time  con- 
tinually increase,  it  is  said  to  move  with  an  accelerated  velocity ; 
and  with  a  retarded  velocity,  when  those  spaces  continually  de- 
crease. If  its  motion  is  so  regulated,  that  it  receives  equal  in- 
crements of  velocity  in  equal  times,  then  it  is  said  to  be  uniform- 
ly accelerated ;  and  uniformly  retarded,  if  the  body  suffers  equal 
decrements  of  velocity  in  equal  times. 

The  space  described  by  a  body  moving  with  uniform  velocity,  in- 
creases in  the  compound  ratio  of  the  time  and  velocity. 

For,  a  body  moving  10  seconds,  at  the  rate  of  40  feet  per  sec- 
ond, will  move  over  10x40,  or  400  feet;  and  let  x  equal  the 
number  of  seconds  for  which  a  body  moves  uniformly,  and  y  the 
number  of  feet  described  in  each  second  ;  then  it  is  evident  that 
xy  will  denote  the  number  of  feet  (i.  e.  the  space)  described  by 
it  in  x  seconds.  In  general,  if  S  be  the  space  described  by  a 
body,  T  the  time  of  its  motion,  expressed  in  seconds,  V  the  uni- 
form velocity  with  which  it  moves,  expressed  by  the  number  of 
feet  described  in  a  second,  then,  S=TxV.* 

This  is  the  fundamental  equation  of  uniform  motion,  from 
which  the  other  equations  may  be  derived  by  the  common  rules 

of  Algebra.      For,  since  S=TxV/.T=5,  and  V=~  :  and  if  5= 

the  space,  t=ihe  time,  v=the  velocity  of  any  other  body,  ex- 
pressed in  the  same  manner ;  then  the  relation  between  S,  T,  V, 
and  s,  t,  v,  may  be  expressed  by  the  following  proportion ; 

S  :  s  : :  Tx  V  :  <xv.f  /.S  ocTxV ;  /.T  oc|,  and  V  <x|. 

If  S  be  given,  then  T  oc^,  and  V  oci 

The  laws  of  Uniform  Motion,  therefore,  are  comprehended  in 
the  following  THEOREMS,  which  are  to  be  treasured  up  in  the 
memory. 

I.  The  SPACE  equals  the  product  of  the  time  into  the  velocity ;  J  or 
(when  different  spaces  are  compared)  the  space  varies  as  the 
product  of  the  time  into  the  velocity. 

II.  The  TIME  equals  the  space  divided  by  the  velocity  ;  or  (when 
different  times  are  compared)  the  time  varies  as  the  space  divi- 
ded by  the  velocity. 

III.  The  VELOCITY  equals  the  space  divided  by  the  time  ;  or  (when 

*  The  young  learner  is  apt  to  be  puzzled  with  such  abstract  expressions  as  "  Space 
equal  to  time  multiplied  into  velocity ;"  but  it  may  be  observed  that  by  velocity  is 
meant  nothing  more  than  the  space  passed  over  in  one  second;  which  may  evidently 
be  so  multiplied  as  to  equal  another  space. 

t  Euc.  V.  7. 

t  This  is  a  concise  mode  of  saying,  The  number  expressing  the  space,  equals  the 
product  of  the  number  expressing  the  time  into  the  number  expressing  the  velocity. 


MECHANICS.  25 

different  velocities  are  compared)  the  velocity  vanes  as  the  space 
divided  by  the  time. 

IV.  When  the  space  is  given,  the  time  varies  inversely  as  the 
velocity. 

13.  QUESTIONS  ON  UNIFORM  MOTIONS. 

1.  A  ball  was  rolled  on  the  ice  with  a  velocity  of  30  feet  per 
second,  and  moved  uniformly  45  seconds :    What  space  did  it  de- 
scribe? Ans.  1350  feet. 

2.  A  steamboat  moved  steadily  across  a  lake  53  miles  wide,  at 
the  rate  of  16  miles  per  hour :  What  time  was  occupied  in  cross- 
ing? Ans.  3T\  hours. 

3.  On  the  supposition  that  the  earth  describes  an  orbit  of  600 
millions  of  miles  in  365£  days,  with  what  velocity  does  it  move 
per  second?  Ans.  19  miles,  nearly. 

4.  Three  planets  describe  orbits  which  are  to  each  other  as  15, 
19,  and  12,  in  times  which  are  as  7,  3,  and  5:  What  are  their 
comparative  velocities  ?  Ans.  225,  665  and  252.* 

14.  The  MOMENTUM  of  a  body  is  its  quantity  of  motion,  and  is  as 
the  product  of  its  quantity  of  matter  and  velocity.  The  quantity  of 
motion,  or  momentum,  of  each  particle  evidently  depends  on  its 
velocity ;  and  therefore  the  momentum  of  the  whole  must  depend 
on  the  same  particles  multiplied  into  the  common  velocity.  By 
velocity  is  understood  the  space  moved  over  in  a  second.  Ac- 
cording to  this  definition,  a  body  at  rest  cannot  be  said  to  have  any 
momentum,  but  it  is  then  said  to  have  an  amount  of  inertia  cor- 
responding to  its  quantity  of  matter,  or  mass.  Inertia  opposes  the 
same  resistance  as  momentum  of  similar  amount. 

Let  M  be  the  momentum  of  a  body ;  Q,  its  quantity  of  matter 
or  weight  expressed  in  pounds;  V  its  velocity  expressed  in  feet; 
and  let  m  be  the  momentum,  q  the  weight,  v  the  velocity,  of  any 
other  body  expressed  in  the  same  manner;  then  the  relation 
between  M,  Q,  V,  and  m,  q,  v,  will  be  expressed  by  the  following 
proportion ; 

M:»i::QxV:?xt;;  .-.  MxQxV;  /.Qoc^andVoc^.  If 
Q  be  given,  MocV;  if  V  be  given,  Mac  Q;  andif  Qocy,  or  Vac^-, 

then  M  will  be  given.     If  V  <*:-y,  or  V  :  v  :  :  q  :  Q,  then  QV=qv 

and  M=7ra. 

The  following  THEOREMS  therefore  comprehend  the  doctrine  of 
Momentum. 

t  Day's  Algebra,  360,  cor.  1. 

4 


26  NATURAL   PHILOSOPHY. 

I.  The  MOMENTUM  equals  the  product  of  the  quantity  of  matte* 
into  the  velocity ;  or  (when  different  momenta  are  compared)  the 
Momentum  varies  as  the  product  of  the  quantity  of  matter  into 
the  velocity. 

II.  The  QUANTITY  OF  MATTER  equals  the  Momentum  divided  by 
the  velocity ;  or  (when  different  masses  are  compared)  the  Quan 
tity  of  Matter  varies  as  the  Momentum  divided  by  the  velocity. 

III.  The  VELOCITY  equals  the  Momentum  divided  by  the  quantity 
of  matter;  or  (when  different  velocities  are  compared)  the  Velocity 
varies  as  the  Momentum  divided  by  the  quantity  of  matter. 

IV.  If  the  quantity  of  matter  is  given,  the  momentum  is  as  the 
velocity. 

V.  If  the  velocity  is  given,  the  momentum  varies  as  the  quantity 
of  matter. 

VI.  If  two  bodies  move  with  velocities  which  are  inversely  as  their 
quantities  of  matter,  they  have  equal  momenta. 

Thus,  if  a  ship  of  100  tons,  sailing  at  the  rate  of  7  knots,  meet 
another  ship  of  50  tons,  sailing  14  knots  per  hour,  they  will  en- 
counter each  other  with  equal  momenta.  This  constitutes  a  fun- 
damental principle  in  the  mechanical  action  of  bodies. 

15.  QUESTIONS  ON  MOMENTUM. 

1.  A  ship  weighing  336,000  Ibs.  is  dashed  against  the  rocks  in 
a  storm,  with  a  velocity  of  16  miles  per  hour:  With  what  mo- 
mentum did  she  strike  ?  Ans.  7,884,800  Ibs. 

2.  On  the  supposition  that  Goliath  of  Gath  presented  an  obsta- 
cle of  350  Ibs.,  and  that  the  stone  hurled  by  David's  sling  weighed 
two  ounces :  With  what  velocity  must  it  have  been  thrown  to 
have  prostrated  the  giant  ? 

Ans.  It  must  have  exceeded  2800  feet  per  second.* 

3.  Wishing  to  know  the  velocity  of  a  musket  ball  weighing  1 
oz.,  I  suspended  after  the  manner  of  a  pendulum,  a  log  of  wood 
weighing  53  Ibs.     The  ball  on  entering  the  log  gave  it  a  motion 
of  2  feet  per  second :  What  was  the  velocity  of  the  ball  ? 

Ans.  1698  feet  per  second.^ 

4.  If  a  comet  moving  at  the  rate  of  1,000,000  miles  per  hour, 
were  to  meet  the  earth  moving  19  miles  per  second :  What  ratio 
will  the  mass  of  the  comet  bear  to  that  of  the  earth,  supposing 
that  they  mutually  destroy  each  other's  motions  ? 

Ans.  1  :  14.6  ;  or  the  comet  must  have  nearly  y'j  as  much  matter 
as  the  earth. 

5.  Two  railway  cars  have  their  quantities  of  matter  as  7  to  3, 
and  their  momenta  as  8  to  5  :   What  are  their  respective  veloci- 
ties ?  Ans.  As  24  to  35,  or  nearly  as  5  to  7. 

*  The  maximum  velocity  of  a  cannon  ball  is  usually  reckoned  2000  feet  per  second. 
t  It  is  to  be  remarked,  that  the  ball  moves  with  the  log,  and  therefore  its  mass  ia 
to  be  added  to  that  of  the  log. 


MECHANICS.  27 

16.  FORCE  is  any  cause  -vhich  moves  or  tends  to  move  a  body,  or 
which  changes  or  tends  to  change  its  motion.     (Art.  2.)     Forces 
can,  for  the  most  part,  be  reduced  to  the  three  following  classes, 
attraction,  repulsion,  and  animal  strength.     Thus,  the  power  of 
the"  waterfall  can  be  traced  to  gravitation  ;  that  of  steam  to  the 
repulsive  energies  of  heat ;  and  that  of  the  horse  and  the  ox  to  ani- 
mal strength.     Forces  are  divided  into  two  kinds,  according  to 
the  manner  in  which  they  act.     If  a  force  acts  instantaneously, 
and  then  ceases,  it  is  called  an  impulsive  force.     A  ball,  sudden- 
ly put  in  motion  by  the  hand  or  any  instrument,  along  a  horizon- 
tal plane,  is  an  instance  of  the  effect  produced  by  an  impulsive 
force.     When  a  force  acts  incessantly,  it  is  called  an  accelerating 
force,  and  is  either  constant  or  variable  ;  constant,  when  the  in- 
crements or  decrements  of  velocity  caused  by  it,  in  equal  succes- 
sive parts  of  time,  are  equal ;  and  variable,  when  the  increments 
or  decrements  of  velocity  thus  produced,  are  unequal.     The  force 
of  gravity  near  the  Earth's  surface,  is  an  example  of  a  constant 
force  ;  for  it  causes  equal  increments  or  decrements  of  velocity 
in  equal  portions  of  time,  not  by  impulses,  but  by  incessant  action. 
Gravity  at  different  distances  from  the  earth,  is  a  variable  force, 
whose  variation  is  estimated  in  the  same  manner  as  that  of  weight 
in  article  8. 

17.  Different  constant  forces  generate  velocities,  which  are  as  the 
product  of  the  forces  and  times. 

Let  T  denote  the  time,  and  F  the  constant  force  ;  and  con- 
ceive the  time  to  be  divided  into  exceedingly  small  equal  por- 
tions ;  then,  since  equal  impulses,  and  of  course  equal  velocities, 
are  added  to  the  moving  body  at  each  of  these  instants,  the  whole 
velocity  acquired,  must  be  proportioned  to  that  of  each  instant 
(which  is  the  measure  of  F)  multiplied  by  the  number  of  in- 
stants;  or  Vac  FxT. 

One  steam  car  was  propelled  by  a  constant  force  of  25  Ibs.  for 
10  minutes,  above  what  was  sufficient  to  overcome  all  resist- 
ances, and  another  was  driven  by  a  similar  force  of  18  Ibs.  for  7 
minutes  :  What  were  their  comparative  velocities  ? 

Ans.  As  250  to  126  ;  or  the  first  car  had  nearly  twice  the  veloci- 
ty of  the  second. 

18.  There  are  three  great  principles  of  motion,  called  the  LAWS 
OF  MOTION,  derived  from  universal  experience  and  observation, 
and  of  such  extensive  application  as  to  comprehend  all  the  phe- 
nomena of  mechanics. 

19.  FIRST  LAW. — A  body  continues  in  the  state  in  which  it  is, 
whether  of  rest  or  motion,  until  compelled  by  some  external  force  to 
change  its  state.     That  a  body  at  rest  will  continue  at  rest,  is  a 
consequence  immediately  arising  from  the  inertia  of  matter. 


28  NATURAL   PHILOSOPHY. 

That  a  body  in  motion  will  continue  to  proceed  uniformly  along 
the  right  line  in  which  it  began  to  move,  until  it  is  acted  upon  by 
some  external  force,  is  inferred  from  the  fact,  that  any  deviation 
from  uniform  rectilinear  motion,  in  a  moving  body,  is  observed  to 
be  owing  to  some  external  force  ;  and  that  such  deviation  is  di- 
minished in  proportion  as  such  external  force  is  withdrawn; 
hence,  were  it  entirely  withdrawn,  we  infer  that  the  motion  of 
the  body  would  then  become  uniform,  rectilinear,  and  perpetual. 
We  may  see  approximations  to  such  a  state  in  a  ball  rolled  suc- 
cessively on  the  earth,  on  a  floor,  and  on  smooth  ice.  The  most 
general  impediments  to  motion  are  friction,  resistance  of  the  air, 
and  gravity.  But  if  a  small  wheel  is  put  in  motion  round  a  hori- 
zontal axis,  the  effect  of  gravity  is  taken  off,  (since  one  side 
of  the  wheel  gains  as  muct  in  falling,  as  the  other  loses  in 
rising,)  and  no  impediments  remain  but  the  resistance  of  the  air 
and  friction,  the  former  of  which  may  be  removed  by  placing  the 
apparatus  in  the  vacuum  of  an  air-pump,  and  the  latter  may  be 
greatly  diminished  by  methods  to  be  described  hereafter.  In 
proportion  as  these  several  impediments  are  removed,  the  wheel 
approximates  to  a  motion  which  is  uniform  and  continued.  A 
pendulum  has  been  constructed  to  move  with  so  little  resistance 
as,  when  barely  set  in  motion  with  the  finger,  to  continue  to  vi- 
brate 24  hours. 

20.  SECOND  LAW. — Motion,  or  change  of  motion,  is  proportioned 
to  the  force  impressed,  and  is  in  the  direction  of  that  force.  It  has 
already  been  observed,  that  every  change  in  the  state  of  rest  or 
motion  in  a  body  must  be  effected  by  the  agency  of  some  force  ; 
this  Second  Law  asserts,  that  this  change  will  in  all  cases  be 
proportional  to  that  force,  and  will  be  produced  in  the  direction 
in  which  the  force  acts. 

That  motion  or  change  of  motion  in  a  body  will  be  proportional 
to  the  force  which  produces  it,  is  also  inferred  from  observation 
and  experiment,  as  well  as  from  the  known  connection  between 
cause  and  effect.  Thus,  a  ball  which  moves  with  a  double  or 
triple  velocity  is  found  to  generate  in  another,  by  impulse,  a 
double  or  triple  velocity.  Two  bodies  meeting  with  equal  quan- 
tities of  motion  mutually  stop  each  other.  Two  forces,  which, 
by  acting  singly  during  equal  times,  produce  equal  velocities  in 
some  third  body,  are  found  by  acting  together  during  the  same 
length  of  time,  to  produce  a  double  velocity.  If  a  new  force  is 
impressed  upon  a  body  in  motion,  in  the  direction  in  which  it 
moves,  its  motion  is  increased  proportionally  to  the  new  force  im- 
pressed :  if  this  force  acts  in  a  direction  contrary  to  that  in  which 
the  body  moves,  it  is  found  to  lose  a  proportional  part  of  its  mo- 
tion :  if  the  direction  of  this  force  is  oblique  to  that  of  the  mov- 
ing body,  it  gives  it  a  new  direction  compounded  of  both.  A 
force  which  we  know  to  act  equally,  produces  equal  increments 


MECHANICS. 


29 


of  velocity  in  equal  times.*     Hence  it  follows,  that  the  smallest 
force  is  capable  of  moving  the  largest  body. 

With  respect  to  the  direction  in  which  a  body  moves,  it  is  evi- 
dent that  when  it  is  under  the  direction  of  any  given  force, 
whether  it  be  an  impulsive  one,  or  one  that  acts  incessantly,  the 
body  can  have  no  tendency  whatever  to  deviate  to  the  one  side 
or  to  the  other,  f  but  must  proceed  along  the  right  line  in  which 
the  force  acts. 

21.  THIRD  LAW. — When  bodies  act  on  each  other,  action  and  re- 
action are  equal  and  in  opposite  directions.  The  meaning  of  this 
law  is,  that  when  a  body  imparts  motion  in  any  direction,  it  loses 
an  equal  quantity  of  its  own  in  the  opposite  direction — that 
when  a  body  receives  a  blow,  it  gives  to  the  striking  body  an 
equal  blow — that  when  A  presses  on  B,  B  returns  to  it  an  equal 

gressure — and  that  when  it  attracts  or  repels  B,  it  receives  from 
the  same  influence  in  the  opposite  direction. 


Fig.  4. 


This  law  is  brought  to  the 
test  of  experiment  by  means 
of  the  apparatus  represented  in 
Fig.  4,  where  A  and  B  are  two 
balls  of  lead  for  example,  sus- 
pended at  C,  by  a  flexible  line, 
by  which  A  may  be  drawn  out 
towards  X,  and  let  fall  upon 
B.  The  velocities  gained  or 
lost  are  indicated  by  the  grad- 
uated arc  XY  ;  and  it  is  found 
that  when  A  falls  on  B,  what- 
ever motion  A  communicates 
to  B,  is  communicated  to  A  in 
the  opposite  direction ;  that  is,  the  same  amount  is  taken  from  A. 
Thus,  if  the  two  bodies  are  equal,  and  A  falls  on  B  at  rest,  they 
will,  after  the  blow,  move  on  together,  with  half  the  velocity  of 
A, — B  having  acquired,  and  A  having  lost  an  equal  amount  of  mo- 
tion. If  A  is  greater  than  B,  still  it  is  found  that  the  momentum 
gained  by  B  (ascertained  by  multiplying  its  mass  by  the  velocity) 
is  precisely  equal  to  the  momentum  lost  by  A ;  and  if  A  meets  B 
with  a  momentum  greater  than  that  of  B,  the  latter  will  deprive 
A  of  a  momentum  equal  to  its  own,  and  return  along  with  A, 
both  bodies  having  a  momentum  equal  to  the  difference  of  their 
momenta  previous  to  collision.  It  is  a  general  Law  of  the  Ma- 
terial World,  that  no  body  loses  motion  in  any  direction,  without 
communicating  an  equal  quantity  to  other  bodies  in  the  same  di- 

*  Gregory's  Mechanics,  I,  9. 

t  According  to  the  principle  of  the  SUFFICIENT  REASON,  there  heing  no  cause  why 
the  body  should  deviate  to  one  side  of  this  line  rather  than  the  other  *  hence  it  will 
*emain  in  it.  (See  Playfair's  Outlines,  I,  4.) 


30  NATURAL   PHILOSOPHY. 

rection  ;  and  conversely,  that  no  body  acquires  motion  in  any  di 
rection,  without  diminishing  the  motion  of  other  bodies  by  an 
equal  quantity  in  that  same  direction.* 

Now,  the  moving  force  by  which  A  communicates  momentum 
to  B,  is  called  the  action  of  A  ;  and  the  tendency  of  B  to  diminish 
the  momentum  of  A,  is  called  the  reaction  of  B.  Since,  there 
fore,  according  to  this  meaning  of  the  words  action  and  reaction, 
the  effect  produced  by  the  action  of  A  is  equal  to  the  effect  pro- 
duced by  the  reaction  of  B,  action  and  reaction  are  said  to  be 
equal  during  the  impact  of  A  upon  B.  That  these  effects  are  pro- 
duced in  "  opposite  directions,"  is  evident  from  the  very  nature  of 
the  case. 

This  law  applies  not  only  to  the  impact  of  bodies,  but  to  every 
case  in  which  one  body  acts  upon  another.  It  holds  good,  not 
only  when  bodies  come  into  actual  contact,  but  when  they  act 
upon  one  another  at  any  distance  whatever.  A  body,  A  for  in- 
stance, is  sustained  by  another  body,  B,  and  both  bodies  remain 
at  rest ;  if  the  pressure  exerted  by  the  two  bodies  were  not  equal, 
it  is  evident  that  some  motion  would  ensue  ;  which  is  contrary  to 
the  supposition.  If  motion  does  ensue,  then  the  case  becomes,  in 
a  great  measure,  analogous  to  that  of  impact ;  and  the  effects  pro- 
duced, estimated  in  a  similar  manner,  are  found  to  observe  the 
«!ame  law.  The  mutual  attractions  of  bodies  are  also  subject  to 
this  law.  Thus,  if  two  equal  magnets,  connected  with  two  equal 
and  similar  pieces  of  cork,  be  made  to  float  upon  the  surface  of 
water,  as  soon  as  they  come  within  the  sphere  of  attraction,  they 
are  observed  to  move  towards  each  other  in  a  right  line,  with  equal 
velocities,  and  consequently  with  equal  momenta ;  and  as  the  re- 
sistance which  each  body  meets  with  from  the  fluid  is  evidently 
the  same,  we  infer  that  their  actions  upon  each  other  are  equal.-\ 

22.  These  fundamental  principles  of  "Mechanics"  rest  on  three 
different  kinds  of  evidence  : — 

1.  They  are  conformable  to  all  experience  and  observation. 

2.  They  are  confirmed  by  various  accurate  experiments. 

3.  The  conclusions  deduced  from  them  have  always  proved  true 
in  fact  without  exception.J  . 

OBSERVATION  and  EXPERIMENT,  then,  constitute  the  basis  of  the 
science  of  Mechanics.  Observation  is  the  close  inspection,  and 
attentive  examination,  of  those  phenomena  which  arise  in  the 
course  of  nature.^  Experiment  is  an  artificial  trial  made  for  the 

*  Playfair's  Outlines,  I,  8. 

t  If  two  unequal  magnets,  placed  upon  pieces  of  cork  similar  to  each  other,  and 
proportional  to  the  respective  magnets,  were  made  to  float  in  the  same  manner, 
they  would  approach  each  other  with  velocities  inversely  proportional  to  the  quan- 
tities of  matter  moved,  and  consequently  with  equal  momenta;  but  this  experiment  is 
liable  to  very  great  inaccuracy,  from  the  different  resistances  which  the  bodies  would 
meet  with. 

t  See  Gregory's  Mechanics,  I,  9.     Atwood  on  Rectilinear  Motbn,  p.  360. 

$  Leslie's  Natural  Philosophy,  I,  2. 


MECHANICS.  31 

purpose  of  learning  the  powers  of  nature,  or  the  properties  of  sub- 
stances. The  most  comprehensive  results  obtained  by  both  these 
methods,  so  far  as  respects  mechanics,  are  expressed  in  the  fore- 
going Laws  of  Motion.  Upon  these,  therefore,  the  science  of  Me- 
chanics is  built.  Algebra  and  Geometry,  called  in  to  the  aid  of 
these  fundamental  principles,  lead  to  the  discovery  of  new  rela- 
tions, and  bring  to  light  a  great  number  of  subordinate  truths,  of 
the  highest  degree  of  practical  utility.  Granting  the  truth  of  the 
Laws  of  Motion,  as  these  subordinate  truths  are  attained  by  prin- 
ciples purely  scientific,  (namely,  those  of  Algebra  and  Geometry, 
and  especially  the  latter,)  they  are  attended  with  the  evidence  of 
demonstration  ;  but  since  the  conclusions  can  be  no  more  certain 
than  the  premises,  we  can  claim  for  the  truths  in  Mechanics  that 
degree  of  evidence  only,  which  results  from  observation  and  ex- 
periment, applied  in  their  greatest  perfection. 

23.  QUESTIONS  ON  THE  PRINCIPLES  OF  MOTION. 

1.  A  bird  of  passage  was  observed  to  fly  with  a  uniform  velo- 
city of  19  feet  per  second :  Over  what  SPACE  would  she  pass  in  24 
hours?  Ans.  310,909  miles. 

2.  A  lame  man  set  out  to  travel  round  the  world.     He  could 
walk  but  two  miles  an  hour  for  seven  hours  out  of  the  twenty 
four.     Provided  he  could  go  forward,  without  impediment,  on  the 
circumference  of  a  great  circle  of  the  globe,  (25,000  miles,)  what 
TIME  would  he  require  to  complete  the  journey  ? 

Ans.  4  years  and  32  5  f  days. 

3.  A  wind  blows  uniformly  from  the  equator  to  the  pole,  (say 
6000  miles,)  in  12  days :  What  is  its  VELOCITY  per  hour? 

Ans.  20f  miles. 

4.  How  much  weight  would  a  rock  that  weighs  ten  tons  (22,400 
Ibs.)  at  the  level  of  the  sea,  lose  if  elevated  to  the  top  of  a  moun- 
tain five  miles  high ?     (Art.  8.)  Ans.  55.8952  Ibs* 

5.  If  the  Earth  were  a  hollow  sphere,  and  if,  through  a  hole 
bored  through  the  center,  a  man  were  let  down  by  a  rope,  would 
the  force  required  to  support  him  be  increased  or  diminished  as 
he  descended  through  the  solid  crust,  and  where  would  it  become 
equal  to  nothing  ?     (See  Art.  9.) 

6.  How  much  would  a  44  pound  shot  weigh  at  the  center  of 
the  earth ;  and  how  much  at  a  point  half  way  from  the  center  to 
the  surface  ?     (See  Art  10.) 

7.  If  a  hole  were  bored  through  the  center  of  the  earth,  and  a 
stone  were  dropped  into  it,  in  what  manner  would  the  stone  move 

*  The  weight  would  be  ascertained,  in  this  case"  by  the  effect  on  a  spring,  (Art. 
10,)  and  not  by  scales,  since  a  counterpoise  wauld  sustain  a  loss  of  weight  in  the 
same  degree  with  the  body  in  question, 


32  NATURAL   PHILOSOPHY. 

in  its  way  to  the  center  and  after  it  reached  the  center  ?     (See 
Arts.  9  and  10.) 

8.  Suppose  the  battering-ram  of  Vespasian  weighed  5760  Ibs., 
and  was  found  sufficient,  when  propelled  with  a  certain  velocity, 
to  demolish  the  walls  of  Jerusalem ;  and  suppose  that  a  32  pound 
cannon-ball,  fired  with  a  velocity  of  2000  feet  per  second,  is  found 
capable  of  doing  the  same  execution :  What  was  the  velocity  of 
the  battering-ram  ?  Ans.  11.11  per  sec. 

9.  Suppose  a  grain  of  light,  moving  at  the  rate  of  192,000  miles 
per  second,  to  impinge  directly  against  a  mass  of  ice  moving  at  the 
rate  of  1.45  feet  per  second :  What  weight  of  ice  would  the  light 
stop ?  Ans.  99877.832  Ibs.*  or  nearly  44£  tons. 

10.  If  a  ball  of  the  same  density  with  the  earth,  T\th  of  a  mile 
in  diameter,  were  placed  at  the  distance  of  T\th  of  a  mile  above 
the  earth :  What  space  would  the  earth  move  through  to  meet  it, 
the  diameter  of  the  earth  being  taken  at  8000  miles  ? 

Ans.  ¥TTtrT|TTTirir  inch,  nearly. 

11.  The  quantity  of  matter  in  the  sun  being  354,000  times  that 
of  the  earth,  and  their  distances  from  each  other  being  96,000,000 
miles :  If  the  two  bodies  were  abandoned  to  their  mutual  attrac- 
tion, where  would  they  meet  ? 

Ans.  271.186  miles  from  the  center  of  the  sun  ;  or  the  sun  would 
advance  so  far  to  meet  the  earth.\ 

12.  Two  men  are  pulling  a  boat  ashore  by  a  rope,  one  at  each 
end,  A  being  in  the  boat  and  B  on  the  shore :  How  will  the  time 
of  bringing  the  boat  ashore  compare  with  the  time  in  which  A 
would  pull  it  ashore  alone,  were  the  other  end  of  the  rope  fixed 
to  an  immovable  post  ?     (See  Art.  21.)J 


CHAPTER  H. 

OF  THE  LAWS  OF  FALLING  BODIES. 

24.  SINCE,  when  a  body  moves  with  a  uniform  velocity  for  a 
given  time,  the  space  described  is  in  proportion  to  that  time  and 
velocity  conjointly,  (Art.  12,)  therefore  if  one  side  of  a  right- 


*  1  lb.  av.  =  7000  grains. 

t  In  this  problem  it  is  supposed  that  all  the  matter  of  each  body  is  collected  in  its 
center,  (Art.  4,)  and  that  their  own  semi-diameters  present  no  obstacle  to  the  approach 
of  their  centers.  * 

I  This  problem  is  intended  merely  to  illustrate  the  doctrine  of  action  and  reaction. 
No  numerical  answer  is  required. 


MECHANICS. 


Fig.  5. 


Fig.  6. 


angled  parallelogram  represents  the  time 
of  a  body's  motion,  and  the  other  the 
uniform  velocity  with  which  it  moves, 
the  parallelogram  itself  (whose  area  is 
equal  to  the  product  of  the  two  sides) 
will  represent  the  space  described  in 
that  time.  Thus,  let  the1  line  AE  be 
divided  into  any  number  of  equal  parts 
in  the  points  B,  C,  D,  &c.,  and  from 
those  points  draw  the  equal  straight 
lines  AF,  BG,  CH,  &c.,  at  right-angles 
to  AE,  and  complete  the  parallelogram  AFLE ;  then  if  AB,  BC, 
CD,  &c.,  represent  equal  successive  portions  of  time,  and  AF, 
BG,  CH,  &c.,  represent  the  uniform  velocity  with  which  a  body 
moves,  then  will  the  parallelograms  AG,  BH,  CK,  &c.,  represent 
the  spaces  described  in  those  equal  portions  of  time,  and  the  par- 
allelogram AFLE  the  whole  space  described  in  the  time  repre- 
sented by  AE.* 

25.  Suppose   now  that   a  body  moves  uniformly  as  before, 
during  the  equal  successive  portions  of  time  represented  by  AB, 
BC,  CD,  &c.,  but  at  the  end  of  each 

portion  of  time  receives  an  increase 
of  velocity ;  for  instance,  during  the 
time  AB  let  it  move  with  a  velocity 
represented  by  AF,  during  the  time 
BC  with  a  velocity  represented  by 
BH,  &c. ;  complete  the  parallelo- 
grams AG,  BK,  CM,  and  DO,  then 
the  space  described  in  the  time  AB 
will  be  represented  by  the  parallelo- 
gram AG,  in  the  time  BC  by  the  par- 
allelogram BK,  &c.,  and  the  whole 
space  described  in  the  time  AE  by  the  irregular  figure  AFOE. 

26.  Let  us  next  suppose  that  a  body  receives  equal  increments 
of  velocity  at  the  end  of  each  successive  portion  of  time,  so  that 

*  Since  the  track  described  by  a  moving  body  is  a  line,  how  (it  may  be  asked)  can 
the  space  be  properly  represented  by  a  superficies  ? 

To  avoid  misconception  on  this  subject,  it  will  be  useful  for  the  young  learner  to 
recur  to  a  few  elementary  principles.  Quantity  (it  will  be  recollected)  is  any  thing 
which  can  be  increased  or  diminished,  or  which  is  capable  of  being  measured.  (Al- 
gebra, Art.  1.)  Thus  time  is  a  quantity,  whose  measure  can  be  expressed  in  hours, 
minutes,  and  seconds.  Velocity  is  a  quantity,  being  measured  by  the  number  of  feet 
passed  over  in  a  second.  But  these  two  quantities  (time  and  velocity)  have  no  per- 
manent representatives  of  their  own,  like  numbers,  which  are  represented  by  the  digits, 
or  like  magnitudes,  which  are  denoted  by  lines,  surfaces,  and  solids.  Hence,  such 
quantities  as  times,  velocities,  and  forces,  are  denoted  by  representatives  borrowed 
from  those  of  magnitude.  In  the  case  before  us,  the  space  described  by  a  moving 
body  is  represented  by  a  parallelogram,  not  because  the  space  actually  described  has 
any  resemblance  to  a  parallelogram,  but  because  a  parallelogram  has  the  same  rela- 
tion to  the  sides  of  which  it  is  the  product,  as  space  has  to  the  two  quantities,  time 


M 


34 


NATURAL   PHILOSOPHY. 


Fig.  7. 


II 


during  the  second  interval  of  time  it  moves  with  twice  the  velo- 
city, during  the  third  interval  with  three  times  the  velocity,  &c., 
then  will  CK=2BG,  DM=3BG,&c.  But  AC=2AB,  AD=3AB,  &c. 

Therefore,  . 

AC  :  CK  : :  AB  :BG  : :  AD  :  DM,  &c. 
Pence,  the  figures  ABG,  ACK,  ADM, 
&c.,  are  similar  triangles ;  and  if  AG,  B 
GK,  KM,  &c.,  be  joined,  AO  will  be 
a  straight  line,  and  the  figure  AFOE,  c 
which  represents  the  space  described 
in  the  time  AE,  will  differ  from  the 
triangle  AOE  only  by  the  sum  of  the 
triangles  AFG,  GHK,  KLM,  MNO, 
which  are  all  equal  to  each  other.       E 


N 


A 


K 


27.  Now  let  the  intervals  of  time  and  the  corresponding  incre- 
ments of  velocity  be  only  half  what  they  were  in  the  former  in- 
stance.   Bisect  AB,  BC,  CD,  &c.,  (Fig.  8,)  in  b,  c,  d,  &c.,  and 
complete  the  parallelograms  as  before,  then  the  figure  which  rep- 
resents  the  space  described  in  the  Apr— i-?  Fig-  8. 
time  AE  will  differ  from  the  triangle 

AOE  by  the  sum  of  the  small  trian- 
gles Ajfc  #FG,  Ghk,  &c.,  which  is 
only  half  the  sum  of  the  triangles 
AFG,  GHK,  KLM,  &c.,  in  the  pre-  C 
ceding  figure.  By  continually  halv- 
ing  these  intervals  of  time  and  the 
corresponding  increments  of  veloci- 
ty, the  figure  AFOE  will  approach 
to  the  form  of  the  triangle  AOE  ;  and  E 
when  they  are  diminished  ad  infinitum,  Af  (which  represents  the 
velocity  with  which  the  body  begins  to  move)  will  be  equal  to  0  ; 
as  will  also  the  sum  of  the  triangles  Afg,  gFG,  &c. ;  the  space 
therefore  described  by  a  body  beginning  to  move  from  rest  by  the 
continued  action  of  a  force  which  generates  equal  increments  of 
velocity  in  equal  times,  will  be  accurately  represented  by  the 
right-angled  triangle  AOE,  one  of  whose  sides  AE  represents  the 
time  of  the  body's  motion,  and  the  other,  OE,  the  last  acquired 
velocity. 

28.  We  have  seen,  (Art.  24,)  that  in  uniform  motion,  the  time, 
velocity,  and  space,  have  the  same  relation  to  each  other,  as  the 

and  velocity,  of  which  it  is  the  product.  An  identity  being  thus  established  between 
the  relations  that  subsist  among  magnitudes,  and  those  that  subsist  among  such 
quantities  as  have  no  representatives  of  then  own,  the  representatives  of  magnitudes 
may  be  substituted  to  denote  the  relations  of  the  other  quantities  ;  and  thus  a  great 
number  of  new  relations  are  frequently  discovered  to  exist  among  those  quantities, 
because  they  are  known  to  exist  among  the  magnitudes,  as  the  lines,  surfaces,  &c., 
which  are  taken  to  represent  them.  It  is  thus  that  Geometry  becomes  a  powerful 
auxiliary  to  Mechanics. 


MECHANICS.  35 

sides  and  area  of  a  right-angled  parallelogram.  Now  if  a  mov- 
ing body  should  retain  all  the  velocity  it  has  already  acquired,  and 
take  on  equal  increments  at  equal  successive  instants,  we  see  by 
the  last  article,  that  the  whole  space  described  would  be  repre- 
sented by  the  sum  of  the  parallelograms  described  successively. 
We  see,  moreover,  that  the  smaller  the  time  is  taken,  the  nearer 
the  whole  space  approaches  to  a  right-angled  triangle.  But  when 
a  body  is  descending  by  the  force  of  gravity,  its  velocity  in- 
creases continually ;  the  instant  is  reduced  to  nothing ;  and  the 
little  triangles  which  denote,  in  the  other  case,  the  difference  be- 
tween the  figure  described  and  that  of  a  right-angled  triangle, 
vanish,  and  leave  the  triangle  as  the  proper  representative  of  the 
space  described.  The  laws  of  variable  motion,  however,  are 
more  perfectly  exhibited  by  means  of  the  calculus,  than  they  can 
be  geometrically.* 

29.  The  spaces  described  by  bodies  falling  from  rest  under  the 
influence  of  gravity,  are  to  each  other  as  the  squares  of  the  times  in 
which  they  are  described,  or  as  the  squares  of  the  last  acquired  ve- 
locities, or  as  the  times  and  last  acquired  velocities  conjointly. ~\ 

For  let  S  be  the  space  described,  V  the  velocity  acquired  by  a 
body  falling  from  rest  for  the  time  T  ;  s  the  space  described,  v 
the  velocity  acquired  at  any  other  period  t,  of  its  fall ;  then,  from 
what  has  already  been  demonstrated,  if  the  ratio  of  T  :  t  be 
represented  by  the  lines  AB,  A6,  and  the  ratio  of  V  :  v  by  the 
lines  BC,  be,  drawn  at  right-angles  to  them,  A 
the  ratio  of  S  :  s  will  be  represented  by  the  \ 
triangles  ABC,  Abe.  Now, 
ABC  :  Abe  : :  AB3  :  Abz ;  or,  as  BC2  :  be* ; 

or,  as  ABxBC  :  Abxbc.-\     Hence, 
S  :  s  : :  T2  :  tz,  or  as  V2  :  v*,  or  as  T x V  :  txv. 

As  equal  increments  of  velocity  are  gen- 
erated in  equal  times,  it  is  farther  evident 
that  the  velocity  acquired  varies  as  the  time : 
the  same  conclusion  may  also  be  deduced 
from  the  similar  triangles  ABC,  Abe ;  for 
BC  :  be : :  AB  :  Aft,  i.  e.  V  :  v  : :  T  :  t. 

Since  the  spaces  described  are  as  the  squares  of  the  times  ;  if 
a  body  falls  from  rest  for  times  which  are  represented  by  the  num- 
bers 1,  2,  3,  4,  5,  &c.,  the  spaces  described  in  those  times  will  be 
as  the  square  numbers,  1,  4,  9,  16,  25,  &c.  ;  and  the  spaces  de- 

*  See  Young,  Elements  of  Mechanics,  art.  108.     Renwick,  p.  46. 

t  The  demonstration  applies  to  any  uniformly  accelerating  force,  as  well  as  to 
gravity  ;  and  hence,  although  "  Falling  Bodies"  are  here  under  particular  considera- 
tion, yet  the  proposition  may  be  predicated  of  all  bodies  urged  by  uniformly  accelera- 
ting or  constant  forces. 

t  For  the  right-angled  triangles  ABC,  Abe,  are  to  each  other  both  as  the  squares 
of  their  homologous  sides,  (by  Euc.  6,  19,)  and  in  the  ratio  of  the  parallelograms  of 
which  they  are  respectively  halves. 


NATURAL   PHILOSOPHY. 


scribed  in  equal  successive  portions  of  time  will  be  as  the  odd 
numbers  1,  3,  5,  7,  9,  &c.,  as  exhibited  in  the  following  table. 


Times. 

Spaces  described. 

Spaces  described  in  equal  successive  portions  of  time. 

1 

2 
3 
4 
5 

&c. 

1 
4 
9 
16 
25 
&c. 

In  1st  portion  of  time 
2d         .... 

....         1 

4     1  —3 

3d 

9—4  —5 

4th  

.     .   16—9  —7 

5th  

.     .  25—16—9 

&c  

.     .         &c  -&c. 

30.  If  a  body  be  projected  perpendicularly  upward,  with  the  ve 
locity  which  it  has  acquired  in  falling  from  any  height,  it  will  rise, 
to  Hie  point  from  which  it  fell,  before  it  begins  to  descend  again. 

As,  in  the  descent  of  a  body,  the  force  of  gravity  generates 
equal  increments  in  equal  times,  so,  in  its  ascent,  equal  portions 
of  velocity  will  be  destroyed  in  equal  times.  The  spaces  de- 
scribed in  equal  successive  parts  of  time,  by  a  body  thus  ascend- 
ing, reckoning  from  the  beginning  of  its  motion,  will  be  the  same 
as  those  stated  in  the  foregoing  table,  but  in  an  inverted  order  : 
thus,  if  the  time  be  divided  into  four  equal  parts,  then  the  spaces 
described  in  the  descent  of  the  body  during  these  equal  times  are 
as  the  numbers  1,  3,  5,  7,  but  in  its  ascent  they  will  be  as  7,  5, 
3,  1  ;  that  is,  the  space  described  in  the  first  portion  of  time,  in 
its  ascent,  will  be  the  same  as  that  described  in  the  last  portion 
of  time,  in  its  descent,  and  so  on,  till  the  body  arrives  at  its 
highest  point. 

31.  The  space  which  a  body  describes  from  rest  in  any  time,  by 
the  action  of  gravity,  is  HALF  that  which  it  would  describe  in  the 
same  time  with  the  last  acquired  velocity  continued  uniformly. 

Let  the  triangle  ABC  represent  the  space  de-  A 
scribed  by  gravity  in  the  time  AB,  and  BC  the 
last  acquired  velocity;  produce  AB  to  D,  making 
BD  equal  to  AB,  and  complete  the  parallelogram 
BCDE  ;  then,  if  a  body  moves  for  the  time  BD 
with  the  uniform  velocity  represented  by  BC,  the  B 
space  described  in  that  time  will  be  represented 
by  the  parallelogram  BCDE,  (Art.  24 ;)  but  the 
triangle  ABC  is  half  the  parallelogram  BCDE ; 
hence  the  space  described  with  the  continually 
increasing  velocity  during  the  time  AB,  is  half  D 
that  which  would  be  described  in  the  same  time  BD,  with  the 
velocity  BC  continued  uniformly. 

Since  the  space  described  by  a  body  falling  from  rest,  is  half 
that  which  it  would  describe  in  the  same  time  with  its  greatest 
velocity  continued  uniformly,  and  since  a  body  projected  perpen- 
dicularly upward  rises  to  the  same  height  as  that  from  which  it 


Fig.  10. 


MECHANICS. 


37 


Fig.  11. 


must  fall  to  acquire  the  velocity  of  projection,  the  whole  space  de- 
scribed by  a  body  projected  perpendicularly  upward,  is  HALF  that 
which  it  would  describe  in  the  same  time  with  its  first  velocity  con- 
tinued uniformly. 

32.  The  space  described  in  any  time  by  a  body  PROJECTED  DOWN- 
WARD with  a  given  velocity,  is  equal  to  the  space  which  would  be  de- 
scribed with  that  velocity  continued  uniformly  for  that  time,  together 
with  the  space  through  which  a  body  would  fall  from  rest  by  the  ac- 
tion of  gravity  in  the  same  time. 

Let  AD  represent  the  given  velocity  of  projection,  and  AB  the 
given  time,  and  complete  the  right-angled  parallelogram  ABED ; 
produce  BE  to  C,  and  let  EC  repre-  A  D 

sent  the  velocity  generated  by  gravi- 
ty in  the  time  AB  or  DE,  and  join 
DC*  Then,  according  to  what  has 
been  said  in  Art.  24,  a  body  moving 
under  the  influence  of  projection 
alone,  with  a  uniform  velocity  repre- 
sented by  AD,  would  describe  the 
parallelogram  ABED  in  the  time  AB  ; 
and,  by  Art.  27,  a  body  falling  from  a 
state  of  rest  during  the  same  time,  so 

as  to  acquire  the  velocity  represented   BE  C 

by  EC,  would  describe  the  triangle  DEC  ;  hence  the  figure  ABCD 
truly  represents  the  joint  effects  of  both  forces,  or  the  whole 
space  described. 

33.  The  space  described  by  a  body  ASCENDING  for  a  given  time,  is 
equal  to  the  difference  between  the  space  which  would  be  described 
by  the  body  moving  uniformly  for  that  time  with  the  velocity  of  pro- 
jection, and  the  space  through  which  a  body  would  fall  from  rest  by 
the  action  of  gravity  in  the  same  time. 

Let  BC  (Fig.  12)  represent  the  given  velocity  of  projection, 
and  AB  the  time  in  which  it  must  fall  from  rest  to  acquire  that 
velocity ;  draw  BC  at  right-angles  to  AB,  and  join  AC,  then  the 


Fig.  12. 


triangle  ABC  will  represent  the  space 
through  which  the  body  must  ascend 
to  lose  all  its  velocity.  (Art.  30.)  In 
AB  take  any  point  6,  and  complete  the 
parallelogram  BCD6 ;  then  will  be  rep- 
resent the  velocity  of  the  body  at  the 
end  of  the  time  Eb  of  its  ascent,  and 
cD  will  represent  the  velocity  destroy- 
ed by  gravity  in  the  same  time.  But 
the  velocity  destroyed  by  gravity  in  any 
time  is  equal  to  the  velocity  generated 
by  gravity  in  the  same  time,  (Art.  30 ;)  hence  the  triangle  CDc 


38  NATURAL   PHILOSOPHY. 

will  represent  the  space  through  which  a  body  would  fall  from 
rest  in  the  time  CD  or  B6.     Now  the  figure  BCc&  represents  the 

re  through  which  the  body  would  ascend  in  the  time  B6,  and 
parallelogram  BCD6  represents  the  space  through  which  a 
body  would  move  in  the  time  B&  with  the  velocity  BC  continued 
uniformly  ;  but  the  figure  BCc&  is  equal  to  the  difference  between 
the  parallelograms  BCD6  and  the  triangle  CDc. 

34.  The  foregoing  investigations  show  the  ratios  between  the 
velocities,  times,  and  spaces  of  falling  bodies  ;  but  in  estimating 
the  actual  motion  of  bodies  descending  or  ascending  by  the  force 
of  gravity,  it  is  necessary  to  have  recourse  to  some  fixed  stand 
ard  of  measurement  of  space  and  velocity.  Now  it  has  been  as- 
certained, by  the  most  accurate  experiments,  that  a  body  falling 
freely*  from  rest  describes  a  space  equal  to  16TV  feet  in  the  first 
second  of  its  fall  ;  and  (Art.  31)  a  body  so  falling  would  acquire 
a  velocity  which,  if  continued  uniformly,  would  carry  it  over 
32|  feet  (that  is,  twice  the  space,)  in  the  same  time.  If,  therefore, 
'n—lG^  feet,  m  will  express  the  space  fallen  through  from  rest 
in  one  second,  and  2m  will  express  the  velocity  per  second  ac- 
quired in  that  time.  Let  S  be  the  space  described  by  the  body 
in  any  other  time  T,  and  V  the  velocity  acquired  ;  then  since  the 
spaces  are  as  the  squares  of  the  times,  we  have, 

f  (1)  S  :m  :  :  T2  :  I2  .-  .........      S=wT2. 


(3)  V2=4wS  /  ...........  V=2v^. 

(4)  1  :  2m  :  :  T  :  V.    (Art.  29)    .-..•..     V=2mT. 


(6)  S=wT2  .-.  .    T=V-. 

m 

1.  A  body  has  been  falling  for  6  seconds  :  What  space  has  it 
fallen  through  in  that  time,  and  what  is  the  velocity  which  it  has 
acquired  ? 

S=mT2=16TVx  36=579  feet. 
V=2wT  =32^x6=193  feet  per  second. 

2.  How  far  must  a  body  fall  to  acquire  a  velocity  of  50  feet  in 
a  second,  and  how  long  will  it  be  in  falling  ? 

8=38.86  feet. 
T=1.55  seconds. 

*  All  bodies  descending  or  ascending  near  the  surface  of  the  earth,  meet  with 
more  or  less  resistance  from  the  air  ;  so  that,  strictly  speaking,  a  body  can  never  be 
said  to  descend  freely  but  in  the  exhausted  receiver  of  an  air-pump.  It  is  in  a 
vacuum  that  a  body  describes  16  1-12  feet  in  a  second  ;  the  conclusions,  therefore, 
deduced  in  this  section,  will  approximate  to  the  truth  only  in  those  cases  where  the 
resistance  of  the  air  bears  little  or  no  proportion  to  the  weight  of  the  body. 

t  These  are  important  formulae,  and  are  to  be  caref  ally  stored  in  the  memory. 


MECHANICS.  39 

3.  A  body  fell  from  the  top  of  a  tower  which  was  150  feet 
high  :     How  long  was  it  in  falling,  and  what  velocity  had  it  ac- 
quired when  it  got  to  the  bottom  ? 

T=3.054  seconds.     V=98.237  feet  per  second. 

4.  A  body  was  projected  perpendicularly  upward  with  a  ve- 
locity of  100  feet  in  a  second :   How  far  would  it  ascend  before 
it  began  to  return  ? 

By  Art.  30,  the  height  to  which  the  body  would  ascend  is 
equal  to  that  through  which  a  body  must  fall  from  rest  to  acquire 
the  velocity  of  projection  ;  here,  therefore, 

V2        10000       30000 
V=100,  and  §=-  =  5-^=-^=  155.44  feet. 

5.  A  body  was  observed  to  fall  for  3  seconds,  and  afterward 
to  move  uniformly  for  2  seconds  along  the  horizon  with  the  velo- 
city which  it  had  acquired  by  its  fall :     What  was  the  whole  space 
described  in  its  perpendicular  and  horizontal  motion? 

The  space  described  in  its  fall  =mT2=  1 6TV  x  9=  144£  feet.  The 
velocity  acquired  =2mT=32ix3=96i  feet  per  second  ;  and  as  it 
moved  along  the  horizon  for  2  seconds  with  this  velocity,  it  must 
in  that  time  have  described  193  feet ;  hence  the  whole  space  de- 
scribed from  the  beginning  of  its  fall  to  the  end  of  its  horizontal 
motion  is  144f+193,  or  337f  feet. 

6.  A  cannon  ball  fired  perpendicularly  upward,  was  gone  10 
seconds,  when  it  returned  to  the  same  place :  How  high  did  it 
rise,  and  what  was  the  velocity  of  projection  ? 

Ans.  Height  402TV  feet.  Velocity  of  projection  160|  feet  per 
second. 

35.  Since  the  spaces  described  in  equal  successive  parts  of  time 
(by  Art.  29,)  are  as  the  odd  numbers  1,  3,  5,  7,  9,  &c.,  and  since 
the  space  described  by  a  body  falling  from  rest  is  in  the  first  sec- 
ond m  feet,  the  space  described  in  successive  seconds  will  be  m, 
3m,  5m,  1m,  9m,  &c.  feet. 

1.  A  body  had  been  falling  for  5  seconds  :  Compare  the  spaces 
described  in  the  third  and.  fifth  seconds  of  its  fall. 

Ans.  The  space  described  in  the  third  second=80T52-  feet ;  the 
space  described  in  the  fifth  second=144f  feet. 

2.  A  body  has  fallen  through  579  feet :     What  was  the  space 
described  by  it  in  the  last  second  ? 

Ans.  It  will  be  found  that  the  body  has  been  falling  6  seconds ; 
therefore,  the  space  described  in  the  last  second  is  176}i  feet. 

36.  The  method  adopted  in  the  last  example  for  finding  the 


40  NATURAL   PHILOSOPHY. 

space  described  by  a  body  in  the  last  second  of  its  fall,  is  only 
applicable  when  the  time  consists  of  a  determinate  number  of 
seconds ;  but  it  is  not  difficult  to  investigate  a  general  expression 
for  the  space  described  in  the  last  n  seconds,  whatever  be  the 
value  of  T.  For  the  space  described  in  T  seconds=»iT2 ;  and  the 
space  described  in  T— n  seconds=mx  (T— n)2=7»T-— 2mnT+mir ; 
hence  the  space  described  in  the  last  n  seconds = 

mT2—  (wzT2— 2mnT+mri*)=2mnT-mrii=m(2nT— n~) ; 
if  n=l,  then  the  space  described  in  the  last  second=m(2T — 1) ; 
which  expression  will  lead  to  the  same  results  as  the  method 
practiced  in  example  1,  and  is  likewise  applicable  in  cases  where 
the  time  does  not  consist  of  any  even  number  of  seconds.  For 
example,  let  the  time  of  falling  be  6^  seconds  ;  then  the  space 
described  in  the  last  second,  namely,  from  5£  to  6^  seconds,  will 
be  16TVx  (13—1)=  193  feet. 

If  it  were  required  to  find  the  space  described  in  the  second 
immediately  previous  to  the  last  n  seconds,  we  have, 

Space  described  in  the  last  n  seconds=/n(2Tn  —  w2)  (A). 
Ditto  in  the  last  (n+1)  seconds=m2T(n+l)  -  (n+1)2  (B). 

Subtract  (A)  from  (B),  then  the  space  described  in  the  second 
immediate))'  previous  to  the  last  n  seconds=/ra(2T  — 2n  —  1). 

1.  What  was  the  space  described  in  the  last  2  seconds  by  a  body 
which  had  fallen  from  the  top  of  a  tower  300  feet  high  ? 

Ans.  The  whole  time  is  found  to  be  4.32  seconds  ;  therefore, 
the  space  fallen  through  in  the  last  2  seconds  is  213.58  feet. 

2.  A  body  has  been  falling  for  91  seconds:   What  was  the 
space  described  in  the  last  second  but  4  of  its  fall  ? 

Ans.  160|  feet. 

37.  To  find  the  space  described  in  a  given  time  by  a  body  pro- 
jected upward  or  downward  with  a  given  velocity.  Let  V  be  the 
given  velocity  with  which  a  body  is  projected  downward,  and  T 
the  time  of  its  motion ;  then  the  space  described  in  the  time  T 
with  the  uniform  velocity  V  will  be  equal  to  TxV,  and  the  space 
through  which  a  body  would  fall  by  gravity  in  the  same  time  is 
TttT2 ;  hence,  from  what  was  shown  in  Art.  32,  the  space  de- 
scribed in  the  time  T  by  a  body  projected  downward  with  the 
velocity  V  is  equal  to  TxV+mT2;  and  applying  the  same  pro- 
cess of  reasoning  to  Art.  33,  the  space  through  which  a  body 
would  ascend  in  the  time  T,  if  projected  upward  with  a  given 
velocity  V,  will  be  equal  to  Tx  V— mT*. 

1.  A  body  is  projected  downward  with  a  velocity  of  30  feet 
in  a  second :  How  far  will  it  fall  in  4  seconds  ? 

Ans.  377i  feet. 

2.  A  body  is  projected  upward  with  a  velocity  of  120  feet  in 
a  second :  How  far  will  it  rise  in  3  seconds  ?     Ans.  215£  feet. 


MECHANICS.  41 

38.  MISCELLANEOUS  EXAMPLES. 

1.  With  what  velocity  must  a  body  be  projected  downward 
from  the  height  («),  that  it  may  describe  it  in  T  seconds  ? 

Let  #=the  velocity  required ;  then  the  space  described  by  a 
body  projected  downward  with  velocity  (x)  in  the  time  (T)  is 

Ttf-fwT2;  hence  T#+mT2=a  /.  x= — ™ — .     For  instance, 

let  a=150  and  T=2,  then  the  velocity  with  which  a  body  must 
be  projected  downward  from  the  top  of  a  tower  whose  height  is 
150  feet,  so  that  it  may  arrive  at  the  bottom  in  two  seconds=42f 
feet  per  second. 

2.  With  what  velocity  must  a  body  be  projected  from  the  top 
of  a  tower  300  feet  high,  to  reach  the  ground  in  4  seconds  ? 

Ans.  lOf  feet  per  sec. 

3.  The  space  described  by  a  heavy  body  in  the  4th  second  of 
its  fall  was  to  the  space  described  in  the  last  second  except  4,  as 
1  to  3  :  What  was  the  whole  space  described  by  the  body  ? 

The  space  described  in  the  4th  second=7m  ;  the  space  described 
in  the  last  second  but  4  =m(2T— 2n-l)=m(2T  —  9),  where  T= 
the  whole  time  of  falling  ;  hence  from  the  question 

7m  :  m(2T—9)  : :  1 : 3,  /.  2T-9=21,  or  T=15  ; 
the  whole  space  described,  therefore,  (=mT2)=16T11  x 225=3618£ 
feet. 

4.  Suppose  at  the  same  instant  that  a  body  begins  to  fall  from 
rest  from  the  point  D,  another  body  is  projected  upward  from  B 
with  a  velocity  which  would  carry  it  to  A :  It  is  required  to  find 
the  point  where  they  would  meet. 

Let  C  be  the  point  where  the  bodies  would  meet ;  and  let  Fig.  13. 
AB=fl,  BD=6,  DC=x ;  then  will  AD=a-b,  AC=a-b+x.  .  A 

Now  the  time  of  descending  through   DC=(^J  ;  and  the 

time  of  ascending  through  BC  (=timedown  AB— time  down      ~ 
JL  ^ 

AC)=(--Y-(a~b+XY'>  but  the  time  d(>wn  DC  must  be 
\mJ       \     m      I 

equal  to  the  time  up  BC  ;  hence  we  have 


'.  (a-b+x=a*-x*,  and  a-b+x=a+x-2(ax) 
.-.  2(ax)^  =6,  or  4ax=ba,  and  x=b— . 


NATURAL   PHILOSOPHY. 


5.  Suppose  a  body  to  have  fallen  from  A  to  B,  (Fig.  14,)  when 
another  body  begins  to  fall  from  rest  at  D :  How  far  will  the 
latter  body  fall  before  it  is  overtaken  by  the  former  ? 

Let  C  be  the  point  where  one  body  overtakes  the  other,  F'1S- 14- 
and  let  AB=a,  ED=b,  DC=z ;  then  AC=a+b+z.  Now 


time   down  DC— |— )  ,  and  time  down  BC=time  down 

t 

;  but  at  the  mo- 
rn    /       \m/ 

ment  when  the  lower  body  is  overtaken, 

Time  down  DC=time  down  BC,or /- )  *=la+b+x\   _  (£!_)* 

\ml       \      m      I        \ml 

-x)^,  and  x+a+2(ax)^=a+b+x, 


•D 


or  2(ax=,.'.x=. 
4a 


39.  QUESTIONS  ON  FALLING  BODIES. 

1.  From  a  black  cloud  a  flash  of  lightning  was  observed,  and 
15  seconds  afterward  it  began  to  rain:  On  the  supposition  that 
the  rain  began  to  fall  on  the  instant  of  the  flash,  what  was  the 
height  of  the  cloud?*  Ans.  3618.75/ee*. 

2.  A  meteoric  stone  fell  upon  a  projecting  stick  of  timber,  with 
a  momentum  which,  from  the  motion  given  to  the  stick,  was  es- 
timated at  18435  pounds.     It  occupied  in  falling,  10  seconds: 
From  what  height  did  it  fall,  and  what  was  the  weight  of  the 
stone  ?  Ans.  Height,  1608L  feet  ;  Weight,  57.31  Ibs. 

3.  A  man  fell  into  a  pit  500  feet  deep :  How  long  was  he  in 
falling,  and  what  velocity  did  he  acquire  ? 

Ans.  T=5.57  seconds;  V=179.17  feet  per  sec. 

4.  Wishing  to  ascertain  the  difference  in  the  depths  of  two 
wells,  I  dropped  a  pebble  into  one  of  them,  and  heard  it  strike 
the  water  in  6  seconds ;  and  then  into  the  other,  and  heard  it 
strike  in  7  seconds :  What  was  the  difference  of  their  depths  ? 

Ans.  209TV  feet. 

5.  An  archer  wishing  to  know  the  height  of  a  tower,  found 
that  an  arrow  sent  to  the  top  of  it,  occupied  8  seconds  in  going 
and  returning  :  What  was  the  height  of  the  tower  ? 

4ns.  251%  feet. 

*  No  allowance  is  here  made  for  the  resistance  of  the  air,  which,  in  fact,  greatly 
retards  the  descent  of  drops  of  rain. 


MECHANICS.  43 

6.  In  what  time  would  a  man  fall  from  a  balloon  three  miles 
high,  and  what  velocity  would  he  acquire  1 

Ans.  T=31.38  seconds,  or  about  half  a  minute. 

V= 1009. 39  feet  per  second,  or  about  half 
the  maximum  velocity  of  a  cannon  ball* 

7.  A  body  having  fallen  for  3£  seconds,  was  afterward  ob 
served  to  move  along  the  horizon  (with  the  velocity  which  it  had 
acquired  in  its  descent)  for  2|  seconds :  What  was  the  whole 
space  described  by  the  body  from  the  beginning  of  its  fall  ? 

Ans.  418?  feet,  very  nearly. 

8.  Through  what  space  would  the  aeronaut  (in  question  6) 
fall  during  the  last  second?  Ans.  993.3  feet. 

9.  A  body  has  fallen  from  the  top  of  a  tower  340  .feet  high  : 
What  was  the  space  described  by  it  in  the  last  three  seconds  ? 

Ans.  298. 957  feet. 

10.  Suppose  a  body  be  projected  downward  with  a  velocity 
of  18  feet  in  a  second :  How  far  will  it  fall  in  15  seconds  ? 

Ans.  3888|  feet. 

11.  A  body  is  projected  upward  with  a  velocity  of  65  feet  in 
a  second :  How  far  will  it  rise  in  2  seconds  ?        Ans.  Q5%feet. 

12-  With  what  velocity  must  a  stone  be  projected  into  a  well 
450  feet  deep,  that  it  may  arrive  at  the  bottom  in  4  seconds  ? 

Ans.  ~V =48%  feet  in  a  second. 

13.  Upon  a  steeple  160  feet  high,  is  a  spire  of  50  feet ;  at  the 
same  instant  that  a  stone  was  let  fall  from  the  top  of  the  steeple, 
another  was  projected  perpendicularly  upward  from  the  bottom 
of  it,  with  a  velocity  sufficient  to  carry  it  to  the  top  of  the  spire  : 
At  what  point  will  these  stones^meet? 

Ans.  30.476  feet  from  the  top  of  the  steeple. 

14.  Upon  the  top  of  a  tower  200  feet  high,  is  placed  a  flag-staff 
of  26  feet ;  a  bullet  islet  fall  from  the  top  of  this  flag-staff;  and 
at  the  instant  of  its  passing  the  bottom  of  it,  a  stone  is  let  fall 
from  a  window  44  feet  from  the  top  of  the  tower :  At  what  dis- 
tance from  the  bottom  of  the  tower  will  the  bullet  overtake  the 
stone?  Ans.  137.385  feet. 


*  In  this  problem  we  have  an  example  of  the  immense  velocity  which  bodies  fall- 
ing  toward  the  earth  from  a  great  height  finally  acquire,  being,  in  the  case  supposed, 
more  than  eleven  miles  per  minute.  To  form  an  idea  of  the  whole  progress  of  a 
very  distant  body  falling  toward  the  earth,  we  must  conceive  of  it  as  at  first  moving 
with  extreme  slowness,  and  as  accelerated  by  very  small  increments  of  velocity. 
For  although,  in  terrestrial  mechanics,  gravity  is  considered  as  a  constant  force,  pro- 
ducing uniform  acceleration,  yet  it  must  be  remembered  that  it  is,  in  fact,  a  variable 
force,  diminishing  as  the  square  of  the  distance  increases ;  and  hence,  that  the  ac- 
celeration which  it  produces  in  a  given  time,  is  not  only  much  less  at  remote  dis- 
tances from  the  earth  than  at  its  surface,  but  that  the  rate  of  acceleration  itself  is 
constantly  increasing  as  we  approach  the  earth.  The  principles,  therefore,  which 
serve  for  estimating  the  time,  velocity,  and  space,  of  bodies  falling  near  the  earth,  as 
in  the  foregoing  examples,  do  not  answer  for  bodies  falling  from  great  distances. 
The  laws  governing  these  are  investigated  by  means  of  the  Calculus.  Since  the  ac- 


44  XATURAL   PHILOSOPHY. 

CHAPTER  IE. 

OF  THE  COMPOSITION  AND  RESOLUTION  OF  MOTION. 

40.  IN  the  two  preceding  Chapters,  we  have  considered  the 
motion  produced  in  bodies  by  the  action  of  only  a  single  force. 
We  now  proceed  to  show  the  manner  in  which  a  body  would 
move  when  acted  upon  at  the  same  time  by  several  forces.     Let 
us  first  consider  the  case  of  a  body  acted  upon  by  two  forces. 

41.  Two  impulses,  which,  when  communicated  separately  to  a  body 
would  make  it  describe  the  adjacent  sides  of  a  PARALLELOGRAM  in  a 
given  time,  will,  when  they  are  communicated  at  the  same  instant, 
cause  it  to  describe  the  diagonal  in  that  time  ;  and  the  motion  in  the 
diagonal  will  be  uniform. 

Suppose  a  body  placed  at  A  to  be  acted  upon  by  two  forces, 
one  of  which  would  cause  it  to  move  uniformly  over  the  line  AB, 
C  Fig.  15.  D 


A  B 

and  the  other  over  the  line  AC  ifi  the  same  given  time,  then  com- 
plete the  parallelogram  ACDB ;  and  if  both  forces  act  at  the 
same  instant  upon  the  body,  it  will,  by  their  joint  action,  move 
uniformly  over  the  diagonal  AD  in  the  same  time  that  it  would 
have  described  either  of  the  sides  AB  or  AC  by  the  forces  acting 
separately.  For  it  is  evident  that  the  force  which  acts  in  the  di- 
rection AB,  can  have  no  tendency  whatever  to  prevent  the  ac- 
cess of  the  body  toward  the  line  CD,  which  is  parallel  to  AB, 
(2d  Law  of  Motion,  Art.  20.)*  When  both  forces  act  together, 

celerating  force  diminishes  so  rapidly  as  the  distance  from  the  earth  increases,  there 
is  a  limit  to  the  velocity  which  a  body  can  acquire  by  gravity.  If  it  falls  from  an  in- 
finite distance,  it  can  acquire  only  a  velocity  of  about  seven  miles  per  second  ;  and" 
half  of  this  is  gained  within  1354  miles  of  the  earth.  Were  a  body  projected  from 
the  earth  with  a  velocity  of  7  miles  per  second,  it  would  never  return.  (See  Vince's 
Fluxions,  Sec.  VIII,  Prop.  XL,  5.  Young's  Elements  of  Mechanics,  Art.  116.) 

*  "  When  any  force  is  exerted  upon  a  body  already  in  motion,  the  motion  which 
the  force  would  produce  upon  a  body  at  rest,  is  compounded  with  the  previous  mo- 
tion in  such  a  way,  that  both  produce  their  full  effects  parallel  to  their  own  direc- 
tions." (Whewell,  228.)  That  any  force  impressed  upon  a  body,  already  moving 
under  the  influence  of  different  forces,  has  its  full  effect,  either  in  producing  or  in 
destroying  motion,  is  evident  from  the  fact  that  a  given  force  is  found  to  have  the 
same  effect  upon  bodies  in  different  parts  of  the  earth,  although,  in  consequence  of  the 
diurnal  motion  of  the  earth,  bodies  apparently  at  rest  are  moving  with  various  veloci- 
ties  in  different  latitudes.  (Ib.  231.) 


MECHANICS. 


45 


therefore,  it  will,  by  the  action  of  the  force  in  the  direction  AC, 
arrive  at  the  line  CD  (but  in  a  different  point  of  that  line)  in  the 
same  time  as  if  the  force  in  the  direction  AB  had  not  acted ;  for 
the  same  reason,  the  force  in  the  direction  AC  will  have  no  ten- 
dency to  prevent  the  access  of  the  body  toward  BD,  which  is 
parallel  to  AC  ;  it  will  arrive  therefore  at  the  line  BD  in  the 
same  time  as  if  the  force  in  the  direction  AC  had  not  acted. 
Hence  the  body  will  arrive  at  the  lines  CD  and  BD  at  the  same 
instant  of  time,  and  consequently  will  be  found  at  their  common 
intersection  D  ;  and  as  the  body,  after  it  leaves  the  point  A,  is 
acted  upon  by  no  external  force,  it  must,  by  the  first  law  of  mo- 
tion, have  described  the  diagonal  AD  with  a  uniform  motion. 

42.  As  this  motion  of  a  body  in  the  diagonal  of  a  parallelo- 
gram by  the  joint  action  of  two  forces  which  (acting  separately) 
would  have  caused  it  to  describe  the  two  sides,  is  a  fundamental 
theorem  with  respect  to  the  composition  of  motion,  let  us  con- 
sider it  in  another  point  of  view.  Let  the  lines  AC,  AB  be  di- 


Fig.  16. 


vided  into  the  same  number  of  small  equal  parts,  Aa,  ab,  be,  &c. ; 
Ac?,  de,  ef,  &c.,  which  will  be  to  each  other  as  the  whole  lines 
AC,  AB,  i.  e.  Aa  :  Ad  : :  AC  :  AB  ;  ab  :  de  : :  AC  :  AB  ;  &c.,  and 
consequently  (Alg.  Art.  388,)  Aa+ab  (Ab)  :  Ad+de  (Ae)  ::  AC  :  AB; 
&c.  If  therefore  the  parallelograms  Adga,  Aekb,  &c.,  be  com- 
pleted, then  (by  Euc.  6,  26,)  the  points  g,  k,  m,  &c.,  will  all  fall 
in  the  diagonal  AD.  Now  since  AC,  AB  are  described  uniformly 
in  the  same  time,  the  proportional  parts  Aa,  Ad ;  Ab,  Ae,  &c., 
will  be  described  uniformly  in  the  same  time.  From  what  has 
already  been  demonstrated  therefore,  at  the  end  of  those  differ- 
ent parts  of  time  the  body  will  be  brought  to  the  point  g,  k,  m, 
&c.,  by  the  united  action  of  the  forces  which  would  have  sepa- 
rately made  it  move  over  Aa,  Ad ;  Ab,  Ae,  &c.  Let  the  number 
of  parts  into  which  AC,  AB  are  divided  be  indefinite,  then  the 
number  of  points  g,  k,  m,  &c.,  will  be  indefinite,  and  the  lines, 
Ag,  gk,  km,  &c.,  will  be  indefinitely  small ;  the  body  therefore 
will  begin  to  move  in  the  line  Ag,  and,  being  found  at  the  end 
of  each  successive  instant  of  time  in  the  line  AD,  it  must  have 


46  JfATURAL   PHILOSOPHY. 

moved  over  that  line  with  the  uniform  velocity  with  which  it 
set  off.* 

43.  If  a  body  be  acted  upon  by  two  forces,  one  of  which  would 
cause  it  to  move  uniformly  over  one  side  of  a  TRIANGLE,  and  the  other 
over  another  side  of  the  triangle,  in  the  same  time,  then  by  the  joint 
action  of  those  forces  it  will  be  made  to  describe  the  third  side,  in  the 
same  time  that  it  ivould  have  described  either  of  the  sides  by  the  forces 
acting  separately. 

Thus  if  a  body  be  acted  upon  by  two  forces,  one  of  which 
would  cause  it  to  move  uniformly  over  the  side  AC,  (Fig.  15,) 
and  the  other  over  the  side  CD,  of  the  triangle  ACD,  (or  over  a 
line  parallel  and  equal  to  CD,)  then,  by  the  joint  action  of  those 
forces,  it  would  be  made  to  describe  the  third  side  AD  in  the 
same  time  that  it  would  have  described  either  of  the  sides  AC, 
CD,  by  the  forces  acting  separately.  For  if  the  parallelogram 
ACDB  be  completed,  then,  since  AB  is  equal  and  parallel  to  CD, 
a  force  acting  in  the  direction  AB  would  make  a  body  describe 
AB  in  the  same  time  as  that  in  which  it  would  describe  CD ; 
but  by  what  has  already  been  proved,  if  two  forces  act  upon  a 
body,  by  one  of  which  it  would  be  made  to  describe  AC,  and  by 
the  other  AB,  in  the  same  time,  it  would  by  the  joint  action  of 
these  forces  be  made  to  describe  the  diagonal  AD,  which  is  the 
third  side  of  the  triangle  ACD. 

44.  Let  us  next  suppose  a  ^       Fig.  17. 
body  placed  at  A,  (Fig.  17,) 

and  acted  upon  by  three  for- 
ces, by  one  of  which  it  would 
be  made  to  describe  AB,  by 
another  AC,  and  by  the  thiid  A, 
AD,  uniformly  in  the  same 
time  ;  complete  the  parallel- 
ogram ABEC,  and  join  AE  ; 
complete  also  the  parallelo- 
gram AEFD,  and  join  AF  ; 
then  AF  will  be  the  line  over 
which  the  body  will  move  uni- 
formly by  the  joint  action  of  those  forces,  in  the  same  time  in 


*  The  body  will  also  describe  the  diagonal  AD,  when  acted  upon  by  uniformly 
accelerating  forces.  For,  let  Ac,  Ad,  (Fig.  16,)  be  two  spaces  described  in  the 
times  T,  t,  by  one  uniformly  accelerating  force,  and  A.b,  Aa,  be  two  spaces  described 
by  another  similar  force,  then, 

Ae  :  Ad  : :  T2  :  <2 

A6  :  Aa  : :  T2  :  <2 

.«.  Ae  :  Ad  : :  Ab  :  Aa ; 

consequently,  the  points  g,  Jc,  which  denote  the  positions  of  the  body  at  the  end  of 
each  successive  instant,  are  all  in  the  same  straight  line.     (Euc.  VI,  26.) 


MECHANICS.  47 

which  it  would  have  described  AB,  AC,  or  AD,  by  either  of  the 
forces  acting  separately.  For  by  Art.  41,  a  body  acted  upon  by 
two  forces  in  directions  AB,  AC,  would  be  made  to  describe  the 
diagonal  AE  ;  a  body  placed  at  A  therefore,  and  acted  upon  by 
three  forces  in  directions  AB,  AC,  AD,  is  under  the  same  circum- 
stances as  if  it  was  acted  upon  by  two  forces,  one  of  which 
would  make  it  describe  AE,  and  the  other  AD  in  the  same  time ; 
but  the  line  over  which  a  body  would  move  uniformly  by  the 
action  of  two  forces  in  the  directions  AE,  AD  is  the  diagonal 
AF ;  AF  therefore  is  the  line  over  which  it  would  move  uni- 
formly by  the  joint  action  of  the  three  forces  in  the  directions 
AB,  AC,  AD. 

Since  BE  is  equal  and  parallel  to  AC,  (Fig.  17,)  and  EF  equal 
and  parallel  to  AD,  it  follows,  (for  the  same  reason  as  in  Art.  43,) 
that  if  a  body  be  acted  upon  by  three  forces,  each  of  which  act- 
ing separately  would  make  it  describe  in  succession  the  three 
sides,  AB,  BE,  EF  of  the  figure  ABEF,  (or  lines  parallel  and 
equal  to  them,)  taken  in  the  order  of  the  letters,  A,  B,  E,  F,  it 
would  by  the  joint  action  of  those  forces  be  made  to  describe  the 
fourth  side  AF  in  the  same  time  that  it  would  have  described 
those  sides  respectively  when  the  forces  act  separately.  And 
since  the  same  mode  of  reasoning  applies  to  a  polygon  of  any 
number  of  sides,  we  have  in  general  the  following  theorem. 

45.  If  a  body  be  impelled  by  any  number  of  forces,  which  acting 
separately,  would,  in  a  given  time,  make  it  describe  all  the  sides  of 
a  POLYGON  except  the  last  side  ;  when  all  these  forces  act  at  the  same 
instant,  it  will  be  made  to  describe  the  remaining  side  in  the  same 
given  time, 

Fig.  18. 
B 

Thus,  if  a  body  be  impelled  by 
any  number  of  forces  which,  act- 
ing separately,  would,  in  a  given 
time,  make  it  describe  each  of  the 
sides  AB,  BC,  CD,  DE  of  the  poly- 
gon ABCDE ;  when  all  those 
forces  act  at  the  same  instant,  it 
will  be  made  to  describe  the  re- 
maining side  AE  in  the  same 
given  time. 


46.  If  all  the  sides  of  a  polygon  except  the  last  represent  the 
quantity  and  direction  of  several  forces,  acting  at  the  same  instant 
upon  a  body,  the  remaining  side  will  represent  the  quantity  and  di- 
rection of  a  force  EQUIVALENT  to  them  all 


48  NATURAL   PHILOSOPHY. 

A  force  is  said  to  be  equivalent  to  any  number  of  forces,  when 
it  will,  singly,  produce  the  same  effect  that  the  others  produce 
jointly  in  any  given  time.  The  single  force  is  frequently  called 
the  resultant,  and  the  forces  that  produce  it  are  called  the  com- 
ponents. 

By  the  second  law  of  motion,  the  space  described  is  propor- 
tional to  the  force  impressed  ;  in  all  these  cases,  therefore,  the 
spaces  respectively  described  by  the  body  will  represent  the 
quantity  and  direction  of  the  forces  by  which  it  is  impelled. 
Thus  (see  Fig.  15,)  if  the  quantity  and  direction  of  two  forces 
be  represented  by  the  two  sides  AB,  AC  of  the  parallelogram 
ACDB,  the  diagonal  AD  will  represent  the  quantity  and  direc- 
tion of  a  force  equivalent  to  them  both  ;  or  if  the  two  sides  AC, 
CD  of  the  triangle  ACD  represent  the  quantity  and  direction  of 
two  forces  acting  at  the  same  time  upon  a  body,  the  third  side 
AD  will  represent  a  force  equivalent  to  them  both.  With  re- 
spect also  to  the  forces  by  which  a  body  is  made  to  describe  the 
sides  AB,  BC,  CD,  DE,  of  the  polygon  ABCDE,  (Fig.  18);  if 
AB,  BC,  CD,  DE,  represent  the  quantity  and  direction  of  several 
forces  acting  at  the  same  instant  upon  a  body,  the  remaining 
side  AE  will  represent  the  quantity  and  direction  of  a  force 
equivalent  to  them  all.* 

47.  Since  the  lines  which  represent  the  proportion  of  the  forces 
in  these  different  figures  are  described  in  the  same  time,  and 
since  the  velocity  of  a  body  is  proportional  to  the  space  described 
in  a  given  time,  these  lines  will  also  represent  the  proportion  of 
the  velocities  with  which  they  are  respectively  described.     Thus 
(Fig.  1 5,)  the  velocity  with  which  the  diagonal  AD  is  described 
is  to  the  velocity  with  which  either  of  the  sides  AC  or  AB  is 
described  as  AD  is  to  AC  or  AB ;  and  in  the  case  of  the  polygon 
in  Art.  45,  the  velocity  with  which  the  side  AE  is  described  is  to 
the  velocity  with  which  either  of  the  sides  AB,  BC,  CD  or  DE  is 
described  as  AE  is  to  AB,  BC,  CD  or  DE. 

48.  Hitherto  the  forces  have  been  supposed  to  be  such  as  by 
their  separate  action  would  produce  uniform  velocities  ;  in  which 
case,  a  body  by  their  joint  action  will  be  made  to  describe  a 
straight  line  Math  a  uniform  velocity.     But  if  two  forces  act  upon 
a  body,  by  one  of  which  it  would  be  made  to  describe  a  straight 
line  with  a  uniform  velocity,  and  by  the  other  with  a  variable 
velocity,  then  the  body,  by  the  united  action  of  those  forces,  will 
neither  describe  a  straight  line,  nor  will  it  move  with  a  uniform 
velocity ;  but  will  describe  with  a  variable  velocity  some  curve 

*  It  will  be  remarked  by  the  learner,  that  several  of  these  forces  acting  in  opposite 
directions,  partly  destroy  one  another,  so  that  AE  represents  merely  the  resultant,  or 
what  remains  after  all  these  mutual  actions. 


MECHANICS. 


49 


Fig.  19. 


line,  the  form  of  which  must  be  determined  from  the  particular 
nature  of  the  'two  forces  which  act  separately  upon  the  body. 
Let  us  take  the  case  of  a  body  projected  obliquely  at  the  earth's 
surface,  on  the  supposition  that  it  meets  with  no  resistance  in  its 
passage  through  the  air.  Conceive  a  body  to  be  projected  from 
the  point  P,  (Fig.  19,)  in  the  direction  PN,  with  such  velocity  as 
would  carry  it  uniformly  over  the  line  N 
PN  in  the  same  time  that  it  would  de- 
scend by  the  force  of  gravity  through 
the  space  PV.  Complete  the  parallelo- 
gram PN  VQ ;  then  for  the  same  reason 
as  in  Art.  41,  the  body  at  the  end  of 
that  given  time  would  be  found  in  the 
point  Q ;  having  described,  not  the  di- 
agonal PQ,  but  some  curve  line  POQ.* 
In  PN  take  any  point  M,  and  let  T,  t 
represent  the  times  of  describing  PN, 
PM  respectively ;  make  PL  equal  to  the 
space  through  which  a  body  would  fall 
by  gravity  in  the  time  t,  and  complete  the 
parallelogram  PMOL ;  then  O  will  be 
the  place  of  the  body  at  the  end  of  the  time  t ;  and  in  the  same 
manner  the  other  points  of  the  curve  POQ  might  be  determined. 
Now  since  PN  is  described  with  a  uniform  velocity, 

PN :  PM  : :  T  :  t  .*.  PN2:  PM2: :  T2:*3. 

NQ :  MO : :  T2:  t3.:  NQ :  MO : :  PN2:  PM2. 

Hence  the  curve  is  such,  that  MO  ocPM2,  which  is  a  well  known 
property  of  the  parabola. f  The  curve  POQ,  therefore,  is  a  pa- 
rabola whose  diameter  is  PV,  ordinate  QV,  and  whose  parameter 


49.  QUESTIONS  ON  THE  COMPOSITION  OP  MOTION. 

1.  A  body  is  acted  upon  at  the  same  time  by  two  forces  which 
are  to  one  another  as  a  :  b,  and  their  directions  are  inclined  to 
each  other  in  the  given  angle  A :  What  is  the  magnitude  of  the 
resultant  ? 


*  For  the  body  in  descending  in  the  direction  of  PV,  recedes  from  the  line  PN 
very  slowly  at  first,  but  faster  and  faster  as  it  proceeds  ;  and  since  the  rate  of  accel- 
eration is  augmented  continually,  the  body  must  be  constantly  drawn  further  and 
further  from  the  direction  PN.  But  a  constant  change  of  direction,  implies  that  the 
path  is  a  curve  line. 

t  Bridge's  Conic  Sec.  Art.  27. 

QV2 

t  By   Cor.   to    Prop.   8   of  parabola,  parameterxPV=QV2,  /.  parameter^  py-. 

(Bridge's  Conic  Sections.) 


50  NATURAL   PHILOSOPHY. 

Let  AC  :  AB  (Fig.  15,)  represent  the  ratio  of  a  :  b,  and  let  BAG 
be  equal  to  the  given  angle  A.  Complete  the  parallelogram 
ABDC,  then  AD  will  represent  the  resultant.  Since  CD=AB, 
AC  :  CD  ::«:&;  and  since  CD  is  parallel  to  AB,  the  angles 
BAC+ACD=180°,  .-.  ACD=180°-  L  A ;  hence  the  problem  is  re- 
duced to  finding  trigonometrically  the  third  side  AD  of  the  tri- 
angle ACD,  in  which  are  given  the  two  sides  AC,  CD,  and  the 
included  angle  ACD. 

Let  AC  :  AB  :  :  2  :  3,  and  L  A =60°,  .-.  ACD=  120°  ;  then* 
5:1::  tan.  30°  :  tan.  (iCAD-CDA)  -1  tan.  30° ;  /.f  log.  tan. 
(iCAD— iCDA)=log.  tan.  30°-log.  5=9.0624694=log.  tan. 6° 35'; 
hence  CAD=36°  35',  and  CDA=23°  25',  and 

sin.  23°  25'  (CDA)  :  sin.  ACD  (120°)  : :  AC  (2)  :  AD=4.36, 
i.  e.  if  two  forces  which  are  to  each  other  as  2  :  3  act  upon  a  body 
at  an  angle  of  60°,  the  resultant  will  be  proportional  to  4.36. 

2.  From  an  island  in  the  Straits  of  Sunda,  we  sailed  S.  E.  b  S. 
(33°  45')  at  the  rate  of  6  miles  an  hour  ;  and  being  carried  by  a 
current,  which  was  running  toward  the  S.  W.  (making  an  angle 
with  the  meridian  of  64°  12'i)  at  the  end  of  four  hours,  we  came 
to  anchor  on  the  coast  of  Java,  and  found  the  said  island  bearing 
due  north :  Required  the  length  of  the  line  actually  described  by 
the  ship,  and  the  velocity  of  the  current  ? 

Ans.  S=26.4  miles. 

V=3.7024  miles  per  hour. 

3.  A  sloop  is  bound  from  the  main  land  of  Africa  to  an  island 
bearing  W.  b  N.  (78°  45')  distant  76  miles,  a  current  setting 
N.  N.  W.  (22°  30')  3  miles  an  hour :  What  is  the  course  to  arrive 
at  the  island  in  the  shortest  time,  supposing  the  sloop  to  sail  at 
the  rate  of  6  knots  per  hour ;  and  what  time  will  she  take  ? 

Ans.  Course  76°  41'  4"  S.— Time  lOh.  40m.  7  see. 

50.  We  may  likewise  find  the  magnitude  of  the  force  com- 
pounded of  any  number  of  forces,  whose  quantities  and  directions 
are  represented  by  the  sides  of  the  given  polygon  ABCDE,  (Fig. 
18.)  For  since  AB,  BC  and  ABC  are  given,  AC  and  BCA  may 
be  found;  but  ACD=BCD-BCA, .-.  AC,  CD,  and  ACD  are 
known,  from  which  AD  and  ADE  may  be  determined  ;  and  in  the 
triangle  ADE  we  have  AD,  DE,  and  ADE, .-.  AE  is  known. 

4.  Three  men  are  pulling  at  a  boat  with  equal  forces  and  in  the 
same  plane.     A  pulls  at  right  angles  to  B,  and  B  at  an  angle  of 
45  degrees  with  C :  In  what  direction  will  the  boat  move,  and 

*  Day's  Trigonom.  Art.  .44.  t  Ib.  Art.  41. 


MECHANICS. 


51 


what  is  the  ratio  of  the  resultant  to  the  sum  of  the  individual 
forces  ? 


Fig.  20. 


Let  the  point  of  application  be  at 
A,  (Fig.  20,)  and  let  AB,  AC,  and 
AD,  represent  respectively  the  mag- 
nitude and  direction  of  A,  B,  and 
C.  Then  it  may  be  shown  that  AF 
is  the  resultant,  which  makes  an 
angle  with  AB  of  80°  16',  and  bears 
to  the  sum  of  A,  B,  and  C,  the  ratio 
of  v/3  to  3. 


RESOLUTION  OF  MOTION. 

51.  A  given  force  may  :-be  resolved  into  an  unlimited  number  of 
others,  acting  in  all  possible  directions. 

Let  AB  represent  the  quantity  and  direction  of  some  given 
force  ;  draw  any  lines  AD,  AC,  and  join  DB,  CB  ;  complete  also 
the  parallelograms  ADBE,  Fig.  21. 

ACBF.  Since  AB  is  the  di- 
agonal of  two  parallelograms 
whose  adjacent  sides  are  re- 
spectively AD,  AE,  and  AC, 
AF,  it  may  (by  Art.  41)  be 
considered  as  the  resultant 
of  two  forces  whose  quanti- 
ties and  directions  are  repre- 
sented either  by  AD,  AE,  or 
AC,  AF,  i.  e.  by  AD,  DB,  or 
AC,  CB.  The  forces  repre- 
sented by  AD,  DB,  or  AC, 
CB,  may  also  be  resolved  into 
other  pairs  of  forces,  and  so  on  without  end. 

52.  Sometimes,  however,  by  the  conditions  of  the  problem,  the 
resolved  forces  are  required  to  make  a  given  angle,  or  to  be  in  a 
certain  ratio,  with  each  other.     The  method  of  solving  cases  of 
this  kind,  may  be  illustrated  by  a  few  examples. 

First,  let  a  given  force  AB,  (Fig.  22,)  be  resolved  into  pairs  of 
forces  which  shall  always  act  at  right  angles  to  each  other. 
Upon  AB  describe  a  semicircle,  ABC,  and  from  the  extremities 
of  the  base  draw  straight  lines  to  meet  in  any  point  of  the  cir- 
cumference. The  sides  of  the  triangle  thus  formed,  will  sever- 
ally contain  right  angles  ;*  and  AC  and  a  line  drawn  from  A 

*  Euc.  Ill,  31. 


52 


NATURAL   PHILOSOPHY. 


parallel  and  equal  to  CB,  will  represent  two  forces  equivalent 
to  the  given  force  AB. 

Fig.  23. 

Fig.  22. 

c  V  \  c 


A  B 

Secondly,  let  the  resolved  forces  be  required  to  make  with 
each  other  any  given  angle.  Upon  AB,  (Fig.  23,)  describe  the 
segment  of  a  circle,  ABC,  containing  an  angle,  the  supplement  of 
the  given  angle,*  and  draw  straight  lines  from  the  extremities 
of  the  base  to  any  point  in  the  circumference. 

Fig.  25. 
D 


Fig.  24. 


M 


Thirdly,  let  the  sum  of  the  resolved  forces  be  required  to  be 
equal  to  a  given  quantity.  Let  MN  (Fig.  24)  be  equal  to  the 
sum  of  the  forces  required  and  AB  be  the  given  force  ;  and  upon 
MN,  as  the  transverse,  and  with  the  points  A  and  B  (equally 
distant  from  M,  N,)  as  foci,  describe  the  ellipse  MCN.  From  A 
and  B  draw  straight  lines  to  any  point  in  the  ellipse,  and  the 
sides  of  the  several  triangles  will  form  the  pairs  6f  forces  re- 
quired.f 

Fourthly,  in  like  manner,  pairs  of  forces  whose  difference  shall 
be  always  equal  to  the  same  constant  quantity,  may  be  found  by 
making  A  and  B  the  foci  of  an  hyperbola,  as  in  Fig.  25,  and 
drawing  straight  lines  from  these  points  to  the  curve.J 

1.  A  given  force  (a)  is  required  to  be  resolved  into  different 
pairs  of  forces  which  shall  act  at  an  angle  of  135°  to  each  other: 
What  is  the  radius  of  the  circle  whose  segment  shall  contain  pairs  of 
the  resolved  forces  ? 

*  Euc.  Ill,  33.         t  Bridge's  Conic  Sect.  Art.  8.         t  Conic  Sect  Art  9. 


MECHANICS. 


53 


Let  AB=«,  and  upon  AB  de- 
scribe the  segment  of  a  circle 
which  shall  contain  an  angle 
ADB  of  45°  :  then  the  radius 


2.  To  determine  the  radius  of 
the  circle  when  AB  (Fig.  26)  is 
required  to  be  resolved  into  pairs 
of  forces  acting  at  any  given 
angle  whose  supplement  is  A. 

Let  AB=a,  ADB^A;  find 
the  center  C,  and  join  C  A,  CB  ; 


Fig.  26. 


then  BC  or  radius^ 


axcos.A 
sin.  2A 


Let  a=lO 
A=40° 


then  radius=7.7786. 


53.  The  most  obvious  consideration  with  respect  to  the  com- 
position of  motion  is,  that  if  two  equal  forces  act  upon  a  body  in 
contrary  directions,  they  will  destroy  each  other's  effects,  and 
the  body  thus  acted  upon  will  remain  at  rest ;  or  if  any  two  forces 
act  upon  a  body  in  the  same  straight  line,  then  the  effect  (or,  in 
other  words,  the  motion)  produced  'Will  be  proportional  to  the 
sum  or  difference  of  those  forces,  according  as  they  act  in  the 
same  or  in  opposite  directions.  But  if  these  forces  act  obliquely 
to  each  other,  then  the  resulting  force  will  be  some  intermediate 
quantity  between  that  sum  and  difference,  the  magnitude  of 
which  will  increase  according  as  the  angle  of  inclination  be- 
tween the  directions  of  these  forces  is  diminished.  For  it  is  evi- 
dent that  the  smaller  the  angle  of  inclination  between  two  forces 
is,  the  more  nearly  will  they  conspire  together,  and  consequently 
the  whole  effect  produced  will  be  greater ;  on  the  contrary,  as 
the  angle  of  inclination  increases,  the  two  forces  will  more 
strongly  oppose  each  other,  their  whole  effect  therefore  will  keep 
diminishing. 

D  Fig.  27.  E 


A  B 

This  latter  conclusion  may  also  be  drawn  from  the  geometrical 
representation  of  the  forces.  Let  two  forces  be  represented  by 
AB,  AD,  or  by  AB,  AC,  of  which  AC=AD;  let  the  angle  DAB 
be  greater  than  the  angle  CAB,  and  complete  the  parallelograms 


64  NATURAL   PHILOSOPHY. 

DABE,  CABF ;  then  since  DAB  is  greater  than  CAB,  its  supple- 
ment ABE  must  be  less  than  ABF,  the  supplement  of  CAB ; 
hence  in  the  triangles  ABE,  ABF,  we  have  AB,  BE  equal  to 
AB,  BF,  and  the  angle  ABF  greater  than  ABE,  .•.  AF  is  greater 
than  AE.*  Let  CAB=0,  then  AF=AC+CF=sum  of  the  forces  ; 
let  CAB=180°,  then  AF=CF— AC=difference  of  the  forces; 
in  all  other  cases,  AF  is  of  some  intermediate  magnitude  between 
AC+CF  and  CF— AC,  and  keeps  increasing  as  the  angle  CAB  is 
diminished. 

54.  In  the  composition  of  forces  which  act  obliquely  on  each 
other,  some  force  is  actually  lost ;  for  the  sum  of  the  forces  before 
they  are  compounded  together  is  represented  by  the  two  sides 
AD,  DE  of  a  triangle,  and  after  composition  by  the  third  side  AE. 
The  contrary  happens  with  respect  to  the  resolution  of  forces  ; 
for  the  two  resolved  forces  being  represented  by  the  two  sides  of 
a  triangle  of  which  the  given  force  is  the  third,  the  absolute 
quantity  of  the  resolved  forces  must  be  greater  than  that  of  the 
given  force. 

Five  sailors  raise  a  weight  by  means  of  five  separate  ropes,  in 
the  same  plane,  connected  with  the  main  rope  that  is  fastened 
to  the  weight  in  the  manner  represented  in  figure  29.  B  pulls  at 
an  angle  with  A  of  20° ;  C  with  B,  at  19° ;  D  with  C,  at  21°  30' ; 
and  E  with  D,  at  25°.  A,  B,  and  C,  pull  with  equal  forces,  and 
D  and  E  with  forces  one  half  greater :  Required  the  magnitude 
and  direction  of  the  resultant,  and  the  loss  of  force  occasioned 
by  the  forces  acting  partly  against  each  other. 

Let  the  sides  of  the  polygon  (Fig.  28)  represent  the  several 
forces  in  magnitude  and  direction,  then  AF  will  be  the  resultant 
Fig.  28.  Fig.  29. 


Euc.  I,  24. 


MECHANICS.  55 

The  angle  at  B=160° ;  at  C,  161°  ;  at  D,  158°  30' ;  at  E,  155°. 
Hence, 

1.  The  resultant  makes  an  angle  with  AB— 46°  33'  10". 

2.  Its  value  is  5.1957,  (that  of  all  the  components  being  6,) 
and  it  falls  between  C  and  D. 

3.  The  loss  of  force  is  .1341,  or  about  TyT  of  the  whole. 

55.  A  body  acted  upon  at  the  same  time  by  three  forces,  represented 
in  quantity  and  direction  by  the  three  sides  of  a  triangle  taken  in 
order,  (or  by  lines  parallel  to  these,)  will  remain  at  rest. 
C  Fig.  30.  D 


A  B 

Since  AD  (Fig.  30)  is  equivalent  to  AB  and  AC,  a  body  placed 
at  A  and  urged  by  AB  and  AC  in  one  direction,  and  by  DA  in  the 
opposite  direction,  would  remain  at  rest.  But  these  three  forces 
correspond  in  magnitude  and  direction  with  the  three  sides  of 
the  triangle  ACD. 

56.  If  a  body  be  kept  at  rest  by  three  forces,  those  three  forces 
will  be  represented  by  the  three  sides  of  a  triangle  formed  by  lines 
drawn  in  their  respective  directions. 

For  suppose  a  body  be  kept  at  rest  by  three  forces,  and  that 
AC,  CD  (Fig.  30)  represent  the  quantities  and  directions  of  two 
of  those  forces,  then  the  compound  force  arising  from  those  two 
forces  will  be  represented  by  the  line  AD  ;  a  third  force,  there- 
fore, represented  in  quantity  and  direction  by  the  line  DA,  equal 
and  opposite  to  AD,  must  exactly  counterbalance  AD  and  keep 
it  at  rest.  Whenever,  therefore,  a  body  is  kept  at  rest  by  three 
forces,  if  a  triangle  be  drawn,  whose  sides  are  respectively  in  the 
directions  of  those  forces,  those  sides  will  represent  the  quantity 
and  direction  of  the  several  forces  thus  acting  upon  the  body. 

57.  The  proportion  of  the  three  forces  which  keep  a  body  at  rest 
will  be  represented  by  the  three  sides  of  any  triangle,  drawn  parallel 
or  perpendicular  to  the  sides  of  the  triangle  which  are  in  the  direc- 
tions of  the  forces. 

For,  let  the  triangle  ABC  be  that  whose  sides  are  drawn  in  the 
direction  of  the  three  forces,  then  the  triangle  a/3y  (whose  sides 
are  parallel  to  AB,  BC,  CA,)  and  the  triangle  abc  (whose  sides 
are  perpendicular  to  AB,  BC,  CA,)  being  each  of  them  simila** 

*  Since  the  sides  of  the  triangle  a/?y  ara  respectively  parallel  to  the  sides  of  the 


56 


to  the  triangle  ABC,  must  have  their  sides  a/3,  /3y,  ya,  or  ab,  be,  ca, 
respectively  proportional  to  the  three  sides  AB,  BC,  CA,  which 
represent  the  quantity  and  direction  of  the  forces  acting  upon  a 
body. 

58.  Any  one  of  the  three  forces  which  keep  a  body  at  rest,  is  as  the 
sine  of  the  angle  included  between  the  other  two. 

For,  (Fig.  31,)  AB  :  BC  : :  sin.  BCA  :  sin.  BAG  ;  BC  :  CA  : : 
sin.  BAC  :  sin.  ABC ;  and  CA :  AB  : :  sin.  ABC  :  sin.  BCA. 

Conversely,  if  a  body  be  acted  upon  by  three  forces,  each  of  which 
varies  as  the  sine  of  the  angle  included  between  the  directions  of  the 
other  two,  it  will  remain  at  rest,  since  the  sines  are  as  the  sides  op- 
posite to  them,  and  when  the  forces  are  proportional  to  these 
sides,  the  body  will  remain  at  rest  by  Art.  55. 

59.  A  body  will  be  kept  at  rest  if  it  be  acted  upon  by  any  number 
of  forces,  which  are  represented  in  quantity  and  direction  by  the 
sides  of  a  polygon  taken  in  order. 

For,  let  a  body  be  acted  upon 
by  any  number  of  forces  repre- 
sented by  the  sides  AB,  BC,  CD, 
DE  of  the  polygon  ABCDE ;  then 
(by  Art.  44,)  these  forces  com- 
pounded together  will  be  repre- 
sented in  quantity  and  direction  by 
the  remaining  side  AE  ;  if,  there- 
fore,  at  the  same  time  that  the  body 
is  acted  upon  by  the  forces  AB, 
BC,  CD,  DE,  it  is  also  acted  upon 
by  another  force  represented  in 
quantity  and  direction  by  EA, 
(equal  and  opposite  to  AE,)  it  will 
remain  at  rest.  The  converse  of  this  proposition  may  also  be 

triangle  ABC,  it  is  evident  that  the  angles  a/?y,  j3ya,  ya|3,  are  respectively  equal  to 
ABC,  BCA,  CAB.  With  respect  to  the  triangle  abc;  since  the  angles  at  D,  E,  F, 
are  ri^At-angles,  we  have  DaF4-DAF  =  180Q,  also  DaF+6ac=180°,  /.  DAF=fcac; 
and  in  the  same  manner  it  appears  that  EBF=a6c,  and  ECD=ac6.  (See  Legendre'a 
Geometry,  III,  209.) 


Fig.  32. 


MECHANICS.  57 

established  by  the  same  mode  of  reasoning  as  that  made  use  of 
in  Art.  56,  viz.  :  If  a  body  be  kept  at  rest  by  any  number  offerees, 
those  forces  will  be  represented,  in  quantity  and  direction,  by  the 
sides  of  a  polygon  formed  by  the  intersection  of  lines  drawn  in  the 
direction  in  which  the  forces  respectively  act. 

1.  A  body  is  acted  upon  by  two  forces  a  and  b,  which  are  at 

right  angles  to  each  other  :  It  is  required  to  find  the  magnitude 

and  direction  of  a  third  force,  which  shall  keep  the  body  at  rest. 

Let  AC=a      }  Complete  the  parallelo-     C  Fig.  33.  D 

AE=b      >  gram   ABDC,  and  join 

C  AB=90°  )  AD  ;  then  the  two  forces 
acting  upon  the  body  may  be  represented 
by  AC,  CD  ;  consequently  DA  (the  third 
side  of  the  triangle  ACD)  will  represent 
the  force  which  shall  keep  the  body  at 
rest.  (Art.  55.) 

Now  DA=A 


for  the  magnitude  of  the  force  ;  and  sine  of  CAD  :  sine  of  CDA 
(-DAB)  ::  CD  :  CA  ::&:«;  /.the  direction  DA  of  the  third 
force  divides  CAB  into  two  angles,  whose  sines  are  to  each  other 

as  a  :  b. 

2.  A  body,  acted  upon  by  two  forces,  is  kept  at  rest  by  a  third 
force  (a),  whose  direction  divides  the  angle  contained  between 
the  directions  of  the  two  former  into  the  given  angles  A  and  B: 
What  is  the  magnitude  of  those  two  forces  ? 

Let  AC,  AB  be  the  two  forces,  (Fig.  30)  ;  complete  the  paral- 
lelogram ABDC,  then  DA  (=a)  is  the  third  force  ;  let  CAD=A, 
DAB=B,  then  ACD=180°-(A+B.)  Now,  _ 

AC  :  AD  (a)  :  :  sin.  CDA  or  DAB  (B)  :  sin.  ACD  (180°  -A+B,) 
&  AB  :  AD  (a)  :  :  sin.  ADB  or  CAD  (A)  :  sin.  ACD  (180°-  A+B.) 
sin.  Bxa  sin.  Bxa 


Hence  AC= 


and  AB= 


sin.  (180° -A+B)     sin.  A+B' 

sin.  Axa      sin.jjx«. 

sin.  (180°— A+B)~sin.  A+B 


60.  It  appears  by  Art.  58,  that  any  one  of  the  three  forces 
which  keep  a  body  at  rest  is  proportional  to  the  sine  of  the  angle 
included  between  the  directions  of  the  two  others.  An  import- 
ant consequence  of  this  truth  is,  that,  of  three  forces  that  keep 
a  body  at  rest,  the  two  components  and  the  resultant  may  sever- 
ally be  represented  by  the  sine  of  the  angle  included  between 
the  directions  of  the  two  others ;  viz.  the  resultant  by  the  sine 
of  the  angle  comprehended  between  the  directions  of  the  two 

8 


58 


NATURAL   PHILOSOPHY. 


components,  and  each  of  the  components  by  the  sine  of  the  angle 
comprehended  between  the  resultant  and  the  other  component. 
Hence  are  derived  the  principles  of  the  composition  and  resolu- 
tion Of  PARALLEL  FORCES. 

Thus,  let  a  body  at  A  be  kept  at  rest  by  three  forces  repre- 
sented in  quantity  and  direction  by  the  sides  of  the  triangle  ADC, 
AD  and  DC  being  the  components  and  AC  the  resultant.  Com- 
plete the  parallelogram  ABCD  ;  produce  the  lines  AB,  AC,  AD 
indefinitely,  and  with  any  radius  describe  the  arc  EFG,  and  from 
the  points  E,  G,  draw  the  sines  EH,  El,  GK. 

From  what  has  been  said  it  will  be  seen,  that  the  force  AB 
may  be  represented  by  the  sine  El,  the  force  AD  by  GK,  and  the 
Fig.  34  E 


force  AC  by  EH.  Suppose  now  that  the  lines  AE,  AF,  AG,  ap- 
proach toward  parallelism  by  making  the  center  A  continually 
recede  from  the  arc  EFG ;  then  that  arc  will  continually  ap- 
proach toward  a  straight  line,  while  the  sines  will  approach  to- 
ward a  coincidence  with  it,  until  finally,  when  the  lines  AE,  AF, 
AG,  become  parallel,  the  sine  EH  will  cross  the  parallels  at  right 
angles,  and  the  sines  El  and  GK  will  form  parts  of  the  same 
straight  line  with  EH.  Hence,  when  the  two  forces  AB  and 
AD  become  parallel,  their  resultant  AC  forms  another  parallel 
with  them  both ;  and  since  the  resultant  is  represented  by  the 
sine  EH,  which,  when  these  lines  become  parallel,  equals  the 
two  sines  El  and  GK,  we  hence  derive  the  following  THEOREMS. 

I.  The  resultant  of  two  parallel  forces  is  in  a  direction  constitu- 
ting another  parallel,  and  is  equal  to  their  sum. 

II.  If  a  straight  line  be  drawn  perpendicular  to  the  directions  of 
these  three  forces,  (viz.  the  two  components  and  their  resultant,)  each 
of  the  components  will  be  represented  by  the  part  of  the  perpendicu- 
lar contained  between  the  directions  of  the  two.  others. 

61.  We  have  thus  far  considered  the  two  parallel  forces  as 
acting  the  same  way :  when  they  are  directed  toward  opposite 
parts,  the  investigation  is  the  same  with  that  in  the  last  article, 
and  the  conclusion  the  same,  except  that  the  resultant  is  equal  to 
the  difference  of  the  two  components.  A  great  number  of  par- 


MECHANICS. 


59 


allel  forces  may  be  compounded  into  a  single  force  equivalent  to 
them  all,  by  proceeding  as  in  Art.  44 ;  that  is,  by  first  finding  the 
resultant  of  two  forces,  and  a  new  resultant  for  that  resultant 
and  one  of  the  remaining  forces,  and  so  on  to  the  last ;  and  any 
single  force  may  be  resolved  into  any  number  of  parallel  forces 
by  a  method  the  reverse  of  this. 

62.  In  estimating  the  effects  produced  by  the  composition  and 
resolution  of  forces,  we  have  hitherto  considered  them  as  acting 
in  the  same  plane ;  we  proceed  now  to  the  solution  of  the  prob- 
lem, by  means  of  which  we  are  enabled  to  determine  the  motion 
of  a  body  resulting  from  the  operation  of  any  number  of  forces 

acting  IN  DIFFERENT  PLANES. 

All  the  forces  which  can  possibly  act  upon  a  body,  may  be  resolved 
into  equivalent  forces  acting  in  the  direction  of  THREE  STRAIGHT 
LINES  OR  AXES,  at  right  angles  to  each  other. 

Let  AK,  AG  be  two  straight  lines  drawn  at  right  angles  to 
each  other  in  the  same  plane,  and  let  AL  be  drawn  at  right 
angles  to  that  plane,  and,  consequently,  at  right  angles  to  each 
of  the  lines  AK,  AG.*  Suppose  AB  to  represent  the  quantity 
and  direction  of  a  force  acting  upon  a  body  at  A  ;  let  fall  the 
perpendicular  BP  upon  the  plane  passing  through  AK,  AG  ;  join 
AP,  and  complete  the  parallelogram  APBC.  From  P  draw  PD 
parallel  to  AK,  and  PE  parallel  to  AG.  Since  AB  is  the  diago- 
nal of  the  parallelogram  APBC,  the  force  represented  by  AB  is 
resolved  into  two  others  AC,  AP,  equivalent  to  it ;  and  since 
AP  is  the  diagonal  of  the  parallelogram  ADPE,  the  two  AD. 
Fig.  35. 

G  R 


AE  are  equivalent  to  AP.  Hence  the  given  force  AB  is  re 
solved  into  three  others,  AC,  AD,  AE,  in  the  direction  of  the 
three  straight  lines  AL,  AG,  AK,  which  are  at  right  angles  to 
each  other,  and  issue  from  the  point  A. 

Produce  LA  to  I,  (Fig.  36,)  GA  to  g,  and  KA  to  k ;  so  that  the 


Euc.  2,  Sup.  Def.  I. 


60 


NATURAL   PHILOSOPHY. 


three  lines  (or  axes)  LI,  Gg,  K/i,  shall  cut  each  other  at  right 
angles  in  the  point  A ;  then  it  is  evident  that  the  directions  of 
all  the  forces  which  can  possibly  act  upon  a  body  at  the  point  A, 
will  fall  within  one  or  other  of  the  eight  solid  angles  formed  by 
the  intersection  of  three  planes  cutting  each  other  at  right  angles, 
and  passing  through  the  axes  LI,  Gg ;  LI,  Kk  ;  Gg-,  Kk,  respect- 
ively ;  and  from  what  has  just  been  shown,  each  of  those  forces 
may  be  resolved  into  three  others,  in  the  directions,* 

AL,  AG,  AK ;  L  G 

AL,  AG,  A&;  \  Fig.  36. 

AL,  Ag,  Ak  ; 

AL,  Ag,  AK ; 
or  Al,  AG,  AK ; 

Al,  AG,  Ak ; 

Al,  Ag,  Ak ; 

Al,  Ag,  AK ; 

according  to  the  solid  an- 
gle in  which  it  is  included. 
Thus,  then,  all  the  forces 
which  can  possibly  act 
upon  a  body  at  the  point 
A,  may  be  resolved  into 
others  acting  along  the 

three  axes  LI,  Gg,  Kk ;  for  the  forces  acting  in  the  directions 
AL,  Al ;  AG,  Ag ;  AK,  Ak,  respectively,  are  merely  forces  act- 
ing in  opposite  directions  in  the  same  straight  line. 

COR.  If  the  sum  of  the  opposite  forces  in  the  direction  of  each 
axis  be  equal  to  one  another,  the  body  will  be  at  rest. 

03.  MISCELLANEOUS  QUESTIONS  ON  THE  COMPOSITION  AND  RESOLUTION 
OF  MOTION. 

1.  Three  men,  equal  in  strength,  undertake  to  pull  down  the 
steeple  of  an  ancient  church.     They  fasten  three  ropes  to  a  ring 
near  the  top,  and,  standing  at  equal  distances  from  the  circular 
base  of  the  steeple,  they  pull  at  equal  angles  of  30°  to  each 
other.     The  ropes  severally  make  an  angle  of  40°  with  the  per- 
pendicular axis  of  the  steeple.     Now  if  a  single  force  of  500  Ibs. 
were  applied  at  right  angles  at  the  same  point,  it  would  be  just 
sufficient  to  overturn  the  steeple  :  Required  the  force  actually 
exerted  by  each  man  ?f  Ans.  284.717  Ibs. 

2.  A  body  at  the  equator  moves,  by  the  diurnal  revolution  of 
the  earth,  about  1000  miles,  and  in  lat.  40°  about  766  miles  per 

*  It  will  be  observed  that  the  first  four  angles  lie  above  and  the  last  four  below  the 
plane  that  passes  through  AK  and  AG. 

t  This  problem  requires  no  resolution  in  different  planes.  As  the  given  force  acts 
at  right  angles  to  the  axis  of  the  steeple,  the  three  forces  may  be  considered  as  first 
acting  in  the  same  horizontal  plane,  and  their  resultant  determined.  This  force  in- 
creased in  the  ratio  of  radius  to  the  sine  of  40°,  gives  the  answer. 


MECHANICS.  61 

hour.  Now,  were  a  wind  to  blow  from  the  equator,  commencing 
with  a  course  directly  north,  and  blowing  with  a  uniform  veloci- 
ty of  60  miles  per  hour,  in  what  DIRECTION  would  it  blow  when 
it  reached  the  latitude  of  40°,  supposing  it  still  to  retain  the 
easterly  motion  it  had  in  common  with  other  bodies  at  the  equa- 
tor ?  Ans.  N.  75°  37'  E. 

3.  If  a  man  were  taken  up  at  the  latitude  of  40°,  and  at  the 
same  instant  set  down  at  the  equator,  in  what  DIRECTION  and 
with  what  velocity  would  he  move  on  the  equator  ? 

Ans.  He  would  move  directly  westward,  at  the  rate  of  234  miles 
per  hour. 

4.  A  ferry  boat  crosses  a  river  f  of  a  mile  broad  in  45  min- 
utes, the  current  running  all  the  way  at  the  rate  of  3  miles  an 
hour :  At  what  ANGLE  with  the  direct  course  must  the  boat  head 
up  the  stream  in  order  to  move  perpendicularly  across  ? 

Ans.  71°  34'. 

5.  The  same  things  being  given,  in  what  RATIO  is  the  force 
required  to  move  the  boat  INCREASED,  in  consequence  of  the  cur- 
rent ?  Ans.  It  is  increased  3.162  times. 

6.  I  shot  an  eagle  that  was  flying  directly  over  my  head.     On 
account  of  its  inertia,  it  retained  some  motion  in  a  horizontal 
direction,  and  therefore  fell,  at  the  end  of  4  seconds,  60  feet  from 
the  place  where  I  stood :     Required  the  nature  of  the  CURVE 
which  the  bird  described  in  its  fall  ? 

Ans.  The  curve  is  a  PARABOLA,  of  which  the  equation*  is 
P  x 2571=  3600  ;  and  consequently  the  parameter  (P)=  13.99 
(See  Art.  48.) 


CHAPTER  IV. 

OF  THE  CENTER  OF  GRAVITY. 

64.  THE  center  of  gravity  of  a  body  is  that  point  about  which,  if 
supported,  all  the  parts  of  a  body  (acted  upon  only  by  the  force  of 
gravity)  balance  each  other  in  any  position. 

In  order  to  ascertain  this  point,  it  will  be  necessary  to  resolve 
a  body  into  its  constituent  parts,  and  then  to  find  two  lines, 
about  each  of  which  (if  supported)  these  parts  will  balance  each 
other  in  all  positions  ;  the  common  intersection  of  those  two  lines 
will  be  the  center  of  gravity  required.  In  bodies  of  a  regular 
form  and  uniform  texture  this  is  very  easily  effected,  but  the  dif- 
ficulty increases  as  the  nature  or  shape  of  the  body  becomes 
more  complex.  We  shall  begin  with  showing  the  method  of 

*  Bridge's  Conic  Sec.  Prop.  8.— Cor. 


62 


NATURAL    PHILOSOPHY. 


finding  the  center  of  gravity  of  a  body,  or  of  a  system  of  bodies, 
in  a  few  familiar  instances. 


METHOD    OF   FINDING    THE    CENTER    OF   GRAVITY   OF   A    BODY   OR    SYSTEM 
OF   BODIES. 

65.  In  regular  plane  -figures,  such  as  squares,  parallelograms 
circles,  polygons  inscribed  in  circles,  fyc.,  the  center  of  gravity  is 
the  same  as  the  center  of  the  figures. 

Fig.  37. 
C  C  C  C 


Let  the  annexed  figures  represent  thin  laminae  of  matter  of  a 
uniform  density,  and  let  them  be  divided  into  two  equal  parts  by 
the  straight  lines  AB,  CD.  Conceive  now  each  of  those  laminae 
to  be  resolved  into  lines  of  particles  equal  and  parallel  to  AB, 
there  will  then  be  the  same  quantity  of  mattter  similarly  disposed 
on  each  side  of  AB ;  if,  therefore,  AB  be  supported,  the  parts 
ACB,  ADB  will  balance  themselves  about  it ;  the  center  of  grav- 
ity will  consequently  be  in  the  line  AB.  For  the  same  reason, 
because  all  lines  drawn  parallel  to  AB  are  bisected  by  CD,  the 
center  of  gravity  will  also  be  in  the  line  CD  ;  it  must  therefore 
be  in  their  common  intersection  G.  In  the  same  manner  it  might 
be  shown,  that  the  center  of  gravity  of  regular  solids,  such  as 
the  cube,  parallelepiped,  cylinder,  sphere,  &c.,  is  the  same  with 
the  center  of  magnitude.  For  each  of  these  solids  might  be  di- 
vided into  two  equal  and  similar  parts  by  planes  passing  through 
it  in  three  different  directions ;  the  intersection  of  two  of  these 
planes  would  be  a  right  line,  and  the  intersection  of  that  line  with 
the  third  plane  would  be  the  center  of  gravity  of  the  solid.  Let 
the  three  first  figures  in  the  preceding  page  represent  respectively 
the  section  of  a  cube,  parallelepiped,  and  sphere,  cut  through  their 
middle  ;  then  may  the  line  CD  represent  the  intersection  of  a 
plane  at  right  angles  to  ACBD,  and  AB  the  intersection  of  a 
third  plane  cutting  these  solids  in  a  similar  manner  ;  the  point  G 
will  therefore  be  the  center  of  gravity.  In  the  cylinder  it  is 
evident  that  the  center  of  gravity  will  be  in  the  point  which 
bisects  its  axis. 

66.  When  a  body  is  supported  by  a  prop  placed  under  its  center 
of  gravity,  the  pressure  is  the  same  whether  the  whole  quantity  of 
matter  is  uniformly  diffused  through  the  space  occupied  by  the  body, 
or  whether  it  is  all  concentrated  in  that  center  of  gravity. 


MECHANICS.  63 

Suppose  now  A  and  B  (Fig.  38)  to  be  two  equal  particles  of 
matter  connected  together  by  the  inflexible  rod  AB  void  of  grav- 
ity ;  bisect  AB  in  G,  then  G  will  be  the  common  center  of  grav- 
ity of  A  and  B ;  for  it  is  evident,  that  if  G  be  supported,  the 

|  Fig.  38. 


A  G  B 

two  particles  will  balance  themselves  about  it.  The  pressure 
upon  G  will  be  equal  to  the  weight  of  the  particles  A  and  B,  and 
this  pressure  does  not  at  all  depend  upon  the  length  of  the  line 
AB  ;  it  will  therefore  be  the  same  whether  the  particles  be  placed 
at  A  and  B,  or  a  particle  equal  to  A+B  be  placed  at  G.  The 
same  may  be  said  with  respect  to  the  particles  A;  B,  C,  D,  &c., 
(in  Fig.  39,)  which  are  disposed  uniformly  along  the  inflexible 

Fig.  39. 
ABODE        FKLMN 


rod  AN  void  of  gravity  ;  viz.  that  the  pressure  of  A  and  N  is  the 
same  as  if  A+N  were  placed  at  G  ;  of  B  and  M  the  same  as  if 
B+M  were  placed  at  G  ;  and  that  the  whole  pressure  of  the  par- 
ticles A,  B,  C,  D,  &c.  is  the  same  as  if  A+B+C+D+E+F,  &c. 
were  placed  at  G.* 

This  reasoning  might  be  extended  to  the  lines  of  particles 
composing  the  laminae  in  Art.  65,  for  the  particles  A,  B,  C,  D,  &c. 
(Fig.  39,)  may  be  increased  in  number  till  they  become  contigu- 
ous to  each  other,  and  the  effect  is  the  same  whether  we  consider 
them  as  connected  together  by  an  inflexible  rod  void  of  gravity, 
or  as  actually  united  together  by  the  power  of  cohesion.  Sup- 
posing CD  therefore  to  be  supported,  (see  figures  in  Art.  65,)  the 
pressure  upon  it  will  be  the  same  as  if  all  the  particles  contained 
in  the  lines  parallel  to  AB  were  incumbent  upon  it  ;  and  suppos- 
ing the  point  G  only  to  be  supported,  the  pressure  will  be  the 
same  as  if  the  particles  thus  collected  in  CD  were  incumbent  upon 
it  ;  the  pressure  of  the  lamina  ACBD  upon  the  center  of  gravity, 
is  therefore  the  same  as  if  all  the  matter  contained  in  it  were  in- 
cumbent upon  G.  The  same  mode  of  demonstration  might  be  ap- 
plied to  the  laminae  composing  the  regular  solid  bodies  in  Art.  65. 

67.  Two  weights  or  pressures  acting  at  the  extremities  of  an  in- 
flexible rod  void  of  gravity,  will  be  in  equilibrio  about  a  given  point, 
when  their  distances  from  that  point  are  to  each  other  inversely  as 
those  weights  or  pressures. 

Let  ABCD,  CDEF,  (Fig.  40,)  represent  the  sections  of  two 

*  We  shall  come  to  the  same  conclusion  by  considering  A  and  B,  &c.  as  parallel 
forces,  and  G  as  their  resultant;  then,  by  Art.  60,  G  will  be  equal  to  their  sum. 


04 


NATURAL   PHILOSOPHY. 


cylinders  of  uniform  density  and  of  the  same  diameter,  whose 

axes  are  MN,  NO  ;  bisect  MN  in  G  and  NO  in  g,  then  will  G,  g, 

be  their  centers  of  gravity.     Let  ABCD  be  suspended  from  the 

Fig.  40. 


A 

M 
B 

i" 

D 

S» 

£ 
O 
F 

K 

G 

N       g 

c 

I ,  b>  the  string  PG  attached  to  its  center  of  gravity  ;  and 
let  CDfsF  be  suspended  in  the  same  manner  from  the  hook  Q, 
by  the  string  Qg ;  let  them  also  be  so  placed  that  their  ends 
may  be  contiguous  to  each  other.  Then  will  ABCD  balance  it- 
self about  G,  and  CDEF  about  g,  so  that  the  two  axes  NM,  NO, 
(after  the  cylinders  are  suspended,)  will  lie  in  the  same  straight 
line  ;  and  the  pressures  upon  G,  g,  will  be  the  same  as  if  the  whole 
weights  of  the  cylinders  were  collected  respectively  in  those 
points.  Suppose  now  the  two  ends  which  are  contiguous  to  each 
other,  to  be  firmly  cemented  together,  so  that  the  two  cylinders 
should  become  one  mass ;  this  will  not  at  all  affect  the  pressures 
upon  G,  g,  but  will  merely  serve  to  connect  those  two  points  to- 
gether in  such  a  manner,  that  the  pressures  upon  them  may  be 
considered  as  acting  at  the  extremities  of  an  inflexible  rod  G^ 
void  of  gravity.  Bisect  the  axis  MO  of  the  whole  cylinder 
ABFE  in  the  point  K,  and  K  will  be  its  center  of  gravity;  let 
the  prop  KL  be  placed  under  K,  and  let  the  two  strings  PG,  Qg, 
by  which  it  is  suspended,  be  removed,  and  the  cylinder  will  then 
balance  itself  about  the  point  K  ;  or  in  other  words,  the  two 
pressures  acting  at  G,  g,  will  be  in  equilibrio  about  that  point. 
It  only  remains,  therefore,  to  find  the  relation  of  KG  to  Kg  ;  now 
MK=£MO,  and  MG-4MN,.-.  MK-MG  (or  KG)-i(MO-MN) 
=iNO;  again,  OK=plO,  and  Og-=iNO, .-.  OK-Og  (or  Kg-) 
=4(MO-NO)=iMN  ;  hence  KG:  Kg::  £NO  :  *MN  : :  NO  : 
MN  : :  cylinder  CDEF  :  cylinder  ABCD.  But  the  pressure  upon 
G  (P)  is  equivalent  to  the  weight  of  the  cylinder  ABCD,  and  the 
pressure  upon  g  (p)  to  that  of  the  cylinder  CDEF  ; 
/.KG  :  Kg  :  :p  :  P,  or  P  : p  : :  Kg  :  KG. 

68.  This  furnishes  us  with  the  method  of  finding  the  common 
center  of  gravity  of  any  number  of  bodies  whatever,  connected 
together  by  inflexible  rods  void  of  gravity. 


MECHANICS.  «>fc 

Let  A,  B,  C,  D,  &c.  be  the  bodies,  and  let  the  centers  of  grav- 
ity of  A  and  B  be  connected  together  by  the  inflexible  line  AB. 

Take  A : B : : BG :  AG,*  or  A+B  :  B  : :  BG+AG  (AB)  :  AG, 
then  will  the  bodies  A  and 
B  balance  themselves  about 
G,  (Art.  67,)  and  conse- 
quently G  will  be  their  com- 
mon center  of  gravity,  (by 
Art.  64  ;)  and  the  three 
first  terms  of  the  above  pro- 
portion being  known,  the 
distance  of  G  from  A  is  thus 
found. 

Next,   let   the  center   of  C 

gravity  of  C  be  connected  with  G  by  the  inflexible  line  CG,  then 
for  the  reason  assigned  in  Art.  66,  the  pressure  upon  G  will  be  the 
same  as  if  a  body  equal  to  A+B  were  placed  at  G  ;  take,  therefore, 

A+B  :  C  : :  Cg :  Gg,  or  A+B+C  :  C  : :  Cg+Gg  (CG)  :  Gg,  then  g 
will  be  the  center  of  gravity  of  A+B  and  C,  and  consequently 
the  common  center  of  gravity  of  the  three  bodies  A,  B,  C. 

Again,  let  the  center  of  gravity  of  D  be  connected  with  g  by 
the  inflexible  line  Dg,  then  the  pressure  upon  g  will  be  the  same 
as  if  A+B+C  were  placed  a,t  g.  Take,  therefore, 

A+B+C:D::DK:£K,or,A+B+C+D:D::DK+gK(%):s-K, 
then  K  will  be  the  center  of  gravity  of  A-f-B+C  and  D,  and 
consequently  the  common  center  of  gravity  of  the  four  bodies 
A,  B,  C,  D  ;  moreover,  the  pressure  upon  K  will  be  the  same  as 
if  A+B+C+D  were  placed  at  K  ;  and  thus  we  might  proceed 
for  any  number  of  bodies. 

It  is  evident  that  the  foregoing  demonstration  does  not  at  all 
depend  upon  the  number  or  weight  of  the  bodies,  or  their  dis- 
tance from  each  other ;  it  rests  merely  on  thp  supposition  that 
their  centers  of  gravity  are  connected  together  by  inflexible  rods 
void  of  gravity.  It  may  therefore  be  applied  to  any  number  of 
particles  of  matter  situated  either  in  the  same  or  in  different 
planes,  and  placed  at  all  possible  distances  from  each  other.  In- 
crease the  number  of  these  particles  till  they  become  contiguous 
to  each  other,  and  for  the  imaginary  line  void  of  gravity  substi- 
tute the  power  of  cohesion,  then  the  system  of  bodies,  A,  B,  C,  D, 
&c.  may  represent  an  irregular  mass  of  compact  matter,  not 
unlike  such  as  are  to  be  met  with  in  the  works  of  nature  or  of 
art ;  and  although  it  may  be  difficult  to  find  the  actual  center  of 

*  Since  A  :  B  ::  BG  :  AG,  by  multiplying  extremes  and  means  AxAG=Bx 
BG ;  i.  e.,  when  two  bodies  are  in  equilibrio,  the  product  of  one  of  the  bodies  into 
its  distance  from  the  center  of  gravity,  is  equal  to  the  product  of  the  other  body  into 
its  distance  from  the  same  center.  These  quantities,  AxAG,  and  BxBG,  therefore, 
express  the  respective  forces  by  which  A  and  B  counteract  each  other's  effects  in 
their  tendency  to  motion  round  G. 

9 


NATURAL   PHILOSOPHY. 


gravity  of  such  a  mass,  yet  the  latter  part  of  our  proposition  still 
remains  true ;  viz.  that  if  this  mass  be  supported,  its  pressure 
downward  will  be  the  same  as  if  the  whole  quantity  of  matter 
contained  in  it  were  concentrated  in  its  center  of  gravity. 

69.  Whatever  be  the  form  or  dimensions  of  a  body  upon  a  plane 
parallel  to  the  horizon,  it  will  remain  at  rest,  if  the  line  drawn  from 
its  center  of  gravity  perpendicular  to  the  horizon,  (called  THE  LINE  OP 
DIRECTION,)  falls  within  its  base. 

Fig.  42. 
ADA  D 


HBFC  FB  CO 

For  let  ABCD  (Fig.  42)  represent  the  section  of  a  body  pass- 
ing through  its  center  of  gravity  G,  and  draw  GF  perpendicular 
to  HO,  the  plane  upon  which  it  stands  ;  then,  since  the  tendency 
of  the  body  to  descend  is  the  same  as  if  its  whole  weight  were 
concentrated  in  G,  it  will  rest  or  fall  according  as  G  is  supported 
or  not ;  or  according  as  F  falls  within  or  without  the  base  BC  ; 
moreover,  the  stability  of  the  body  will  depend  upon  the  distance 
at  which  the  point  F  falls  within  the  base. 


Fig.  43. 


70.  If  a  body  be  suspended  freely  from  any  point,  it  will  not  rest 
till  the  line  which  joins  the  center  of  gravity  and  the  point  of  sus- 
pension, is  perpendicular  to  the  horizon. 

For  let  ABCD  Represent  the  sec- 
tion of  a  body  as  before,  G  its  center 
of  gravity,  S  the  point  of  suspension ; 
join  SG,  and  draw  SOW  perpendicular 
to  the  horizon  ;  produce  SG  to  N,  and 
draw  GR  parallel  to  SW  ;  then,  since 
the  weight  of  the  body  may  be  con- 
sidered as  collected  in  G,  its  tendency 
to  motion  will  be  along  the  line  GR. 
Let  GR  therefore  represent  this  ten- 
dency, which  resolve  into  GN  in  the 
direction  SG,  and  RN  perpendicular  to 
it ;  the  part  GN  is  counteracted  by  the 
reaction  from  the  point  of  suspension 
S,  and  NR  is  employed  in  producing 
motion  in  the  direction  of  the  circular 
flfc  GO ;  G  therefore  (and  consequently  the  body)  will  not  re- 


MECHANICS. 


67 


Fig.  44. 


main  at  rest  till  NR  vanishes,  i.  e.  till  the  angle  NGR  (=OSG) 
vanishes,  or  SG  coincides  with  SO. 

Hence  it  follows,  that  if  a  body  be  suspended  successively  by 
different  points,  and  perpendiculars  to  the  horizon  be  drawn 
through  the  points  of  suspension,  the  center  of  gravity  will  lie  in 
each  of  these  perpendiculars,  and  consequently  in  the  point  of 
their  intersection. 

We  proceed  to  apply  the  principles  just  now  investigated  to 
the  solution  of  a  few  practical  examples. 

71.  In  a  TRIANGLE,  if  a  line  be  drawn  from  one  of  the  angles  bisect- 
ing the  opposite  side,  the  center  of  gravity  of  the  triangle  is  in  that 
line  at  the  distance  of  ^  of  its  length  from  the  base* 

Bisect  the  side  'AC  in  D,  and  join 
BD,  which  will  bisect  all  lines  drawn 
parallel  to  AC  ;  consequently,  if  BD  be 
supported,  the  parts  ABD,  DBC  of  the 
triangle  ABC  will  balance  themselves 
on  each  side  of  it  ;  hence  the  center  of 
gravity  is  in  the  line  BD.  Bisect  the 
side  BC  in  E,  and  join  AE  ;  then,  for  A- 
the  same  reason  as  before,  the  center 
of  gravity  will  be  somewhere  in  the 
line  AE  ;  it  must  therefore  be  at  their 
common  intersection  G.  Produce  now  BD  to  F,  and  draw  CF 
parallel  to  EA  ;  then  since  BE=EC,  BG  will  be  equal  toGF; 
but  the  two  triangles  AGD,  DFC,  have  one  side  and  two  angles 
equal  ;  .-.  GD=DF,  and  consequently  GF  (or  BG)=2GD  ;  hence 
BG=|BD,  and  GD=^BD. 

72.  In  a  TRAPEZOID,  the  center  of  gravity  is  i^jfhc  line  that  bisects 
the  two  opposite  sides. 

Let  ABCD  (Fig.  45)  be  a  Fig.  45. 

trapezoid,  and  bisect  AD,  BC,  A 
in  E,  F,  and  join  EF  ;  then 
since  EF  bisects  AD,  BC,  it 
will  bisect  all  lines  drawn 
parallel  to  BC,f  and,  conse- 
quently, the  center  of  gravity 
of  the  trapezoid  is  in  the  line 
EF.  Join  BE,  BD,  DF,  and 
take  GE=iBE, 


then  G  is  the  center  of  gravity  of  the  triangle  ABD,  (Art.  71,) 

*  In  finding  the  centers  of  gravity  of  plane  figures,  a  lamina  of  matter  of  uniform 
density,  in  the  shape  of  those  figures,  is  of  course  understood. 

t  For  if  BA,  FE,  CD  be  produced,  they  will  meet  in  the  same  point,  which  will  be 
the  vertex  of  a  triangle  whose  base  is  BC  ;  and  since  EF  bisects  the  base  BC,  it  will 
bisect  all  lines  drawn  parallel  to  it 


68  NATURAL   PHILOSOPHY. 

and  g  the  center  of  gravity  of  the  triangle  BDC.  Join  Gg; 
then,  conceiving  the  triangles  ABD,  BDC  to  be  collected  in  G,,^, 
their  common  center  of  gravity  must  be  in  the  line  Gg  ;  i.  e.  the 
center  of  gravity  of  the  trapezoid  ABCD  must  be  in  the  line  Gg; 
it  is  also  in  the  line  EF  ;  consequently  it  is  in  K,  the  intersection 
of  EF  and  Gg.  Draw  Gm,  gn,  parallel  to  AD  or  BC  ;  then  since 
EG='BE,  Em  must  be  equal  to  |EF  ;  and  for  the  same  reason 
Fw=jEF ;  .'.  Em—  mn—nF.  Now  K  being  the  common  center 
of  gravity  of  the  triangles  ABD,  BDC, 

GK  :  Kg : :  BDC  :  ABD  : :  BC  :  AD. 

GK:K£-::Km:Kra 
.•.Km:Krc::BC:  AD. 

Km  :  Km+Kw  : :  BC  :  BC+AD. 
And  since  Km+Kn=mn=Em 

.-.Km+Em  :  mn  : :  2BC+AD  :  BC+AD 

EK  :  mn : :  2BC+AD  :  BC+AD  (1) 

Km+Kw  :  Kn  : :  BC+AD  .  AD. 
And  since  mn=nF,  and  Kn+nF=FK 

.'.mn  :  Kn+wF  ::  BC+AD  :  BC+2AD 

mn  :  FK  ::  BC+AD  :  BC+2AD  (2) 
uniting  (1)  and  (2) 

EK  :  FK  ::  2BC+AD  :  2AD+BC. 

If,  therefore,  the  line  EF  be  divided  in  the  ratio  of  the  two  last 
terms  of  this  proportion,  (formed  of  the  known  sides  of  the  trape- 
zoid,) it  will  give  the  center  of  gravity. 

When  AD=0,  then  the  figure  becomes  a  triangle,  and 
EK  :  FK : :  2BC  :  BC  ;  that  is,  FK=|EF,  as  was  found  by  a  dif- 
ferent process  in  Art.  71. 

When  AD=BC,  the  figure  becomes  a  parallelogram,  and 
EK  :  FK  : :  3BC  :  3BC  ;  consequently,  the  center  of  gravity  is  in 
the  center  of  the^gure,  as  was  shown  in  Art.  65. 

73.  The  center  q[  gravity  of  a  POLYGON  may  be  found  by  dividing 
the  polygon  into  triangles,  and  finding  the  common  center  of  gravity 
of  these. 

Let  ABCDEF  be   an  ir-  A         Fig.  46. 

regular  polygon,  divided  into 
triangles  whose  areas  are 
represented  by  P,  Q,  R,  S, 
and  whose  centers  of  gravi- 
ty are  respectively  a,  b,  c,  d. 
Conceive  these  triangles  to 
be  collected  in  the  points 
a,  b,  c,  d  ;  join  «&,  and  take 
bG  :  aG  : :  P  :  Q,  then  G  will 
be  the  center  of  gravity  of 
the  figure  ABCD.  Join  Gc, 
and  take  cg:Gg.:  P+Q,  :  R  ;  then  g  will  be  the  center  of  grav- 
ity of  the  figure  ABCDE.  Let  g  and  d  be  joined,  and  make 


MECHANICS.  69 

dK  :  gK  : :  P+Q+R  :  S,  then  K  will  be  the  center  of  gravity  of 
the  whole  polygon  ;  and  so  we  might  proceed,  whatever  be  the 
number  of  sides. 

If  it  were  required  to  find  the  center  of  gravity  of  the  perime- 
ter of  the  polygon  ;  then  bisect  the  sides  in  the  points  a,  b,  c,  &c., 
(Fig.  47,)  and  (since  the  center  of  gravity  of  a  right  line  is  in  its 
middle  point)  a,  b,  c,  &c.,  will  be  the  centers  of  gravity  of 
the  sides  AB,  BC,  CD,  &c.,  respectively.  Join  ab,  and  take 
bG  :  aG  : :  AB ":  BC  ;  then  G  Fig.  47. 

would  be  the  center  of  grav- 
ity of  that  part  of  the  pe-  y^**""^-^ 
rimeter  represented  by  ABC. 
Again,  join  Gc,   and   take 


then  g  is  the  center  of  grav- 
ity of  such  part  of  the  pe-  g 
rimeter  as  is  represented  by 
ABCD ;  and  so  we  might 
proceed  till  we  had  found 
the  center  of  gravity  of  the 
whole  perimeter. 

74.  The  distance  from  any  assumed  point  of  the  common  center 
of  gravity,  of  any  number  of  bodies  which  have  their  centers  of 
gravity  in  a  right  line  passing  through  that  point,  is  equal  to  the 
sum  of  the  products  arising  from  multiplying  each  body  into  its  dis- 
tance from  the  assumed  point,  divided  by  the  sum  of  the  bodies. 

Let  the  bodies  A,  B,  C,  D,  be  so  placed,  that  the  line  OD  may 
pass  through  their  respective  centers  of  gravity ;  it  is  required 
to  find  the  distance  of  their  common  center  of  gravity  from  any 
point  O,  in  the  line  OD. 

Fig.  48. 
O  A  B  C  D 


G 

Suppose  OD  to  be  an  inflexible  line  void  of  gravity,  and  let  G 
be  the  common  center  of  gravity  of  the  bodies ;  then,  if  G  be 
supported,  the  effort  of  each  body  to  produce  motion  round  G 
would  be  measured  by  the  product  of  its  weight  into  its  distance 
from  G,  (Art.  67  ;)  i.  e.  the  effort  of  A=AxAG ;  of  B=BxBG, 
&c. ;  and  as  the  bodies  are  supposed  to  be  in  equilibrio  about  G, 
the  sums  of  their  efforts  on  each  side  of  G  must  be  equal  to  each 
other,  or 

AxAG+BxBG=CxCG+DxDG,  i.  e. 
A  x  (OG-OA)+B  x  (OG-OB)=C  x  (OC-OG)+D  x(OD-OG) 
.-.AxOG+BxOG+CxOG+DxOG=AxOA+BxOB+CxOC 

+DXOD 


NATURAL    PI'lLOSOPHr. 


Hence  OG= 


A  xOA+B  xOB+C  xOC+D  xOD 
A+B+C+D 


75.  MISCELLANEOUS  EXAMPLES. 

ji 

1.  Three  bodies,  A,  B,  C,  weighing  respectively  3,  2,  and  1 
pounds,  have  their  centers  of  gravity  joined  by  the  lines  AB,  BC, 
CA;  of  which  AB=5  feet,  BC=4,  CA=2 :  What  is  the  distance 
of  their  common  center  of  gravity  from  the  body  C  ? 

Since  the  three  sides  of  the  triangle 
ABC  are  5,  4,  2,  the  three  angles  A,  B, 
C,  will  be  found,  by  the  rules  of  trigo- 
nometry, to  be  respectively  49°  27£', 
22°  20',  and  108°  12>  Let  G  be  the 
center  of  gravity  of  the  bodies  A  and  B, 
and  g  the  common  center  of  gravity  of 
the  three  bodies  found  as  in  Art.  67 ;  then,  since  AB— 5,  A  =3, 
J3=2,  and  A  :  B  : : BG :  AG,  AG  will  be  equal  to  2  feet,  and  BG 
to  3  feet;  hence  in  the  triangle  GAG  there  is  given  AC  =2, 
AG=2,  and  the  angle  CAG=49°  27*',.-.  each  of  the  angles  AGC, 
ACG=65°  161',  from  which  CG  is  found  to  be  equal  to  1.673. 

But  Cg  :  Gg : :  A+B  :  C, 
/.  C#  :  Cg+Gg  (=CG=1.673) : :  A+B  (5)  :  A+B+C  (6); 

hence  Cg-=L67|X5  =1.394  feet. 


50. 


2.  A  cylindrical  tower,  consisting  of  uniform  materials  closely 
cemented  together,  is  20  feet  high,  and  the  diameter  of  its  base 
is  four  feet :  How  far  may  it  deviate  from  its  perpendicular  posi- 
tion, before  it  is  in  danger  of  falling  ? 

Let  ABCD  represent  a  seetion  of  the  tower 
passing  through  its  axis  EF,  and  let  G  be  its 
center  of  gravity.  Suppose  it  to  be  so  much 
inclined,  that  the  perpendicular  line  GB,  let  fall 
from  G,  falls  upon  the  edge  of  its  base  BC ;  then 
GF  (10) :  BF  (2) : :  radius  (1) :  cos.  GFB ;  .'.  cos. 

GFB=^p.200=cos.  78°  27'.    An  angle  of  78° 

27'  is  therefore  the  limit  of  its  inclination,  be- 
fore it  is  in  danger  of  falling.  (Art.  69.)  If 
the  angle  GFB  is  less  than  78°  27'  then  the  per- 
pendicular GB  falls  without  the  base,  and  the 
tower  cannot  sustain  itself. 


MECHANICS.  71 

3.  A  piece  of  timber  of  uniform  density  and  prismatic  form, 
a  section  of  which  perpendicular  to  its  sides,  and  passing  through 
its  center  of  gravity  G  is  represented  by  the  square  ABCD,  is 
placed  upon  an  inclined  plane :  It  is  required  to  show  when  it 
will  have  a  tendency  to  roll,  and  when  to  slide  down  the  plane. 

iTraw  GF  perpendicular  to  the  plane,  -fig.  51. 

and  GK  perpendicular  to  its  base,  and 
let  PLN  be  greater  and  pLN  less  than 
half  a  right  angle.  In  the  former  case, 
since  ELK  is  greater  than  45°,  LEK  or 
GEF  will  be  less  than  45°;  .-.  the  angle 
EGF  is  greater  than  the  angle  GEF 
and  consequently  EF  is  greater  than 
GF  or  BF ;  hence  the  body  has  always 
a  tendency  to  fall  over  in  the  direction 
GE,  and  will  therefore  roll  down  the 
plane  PL.  In  the  latter  case  the  angle  L  K  K  N 

EGF  is  less  than  GEF,  .'.  EF  is  less  than  GF  or  BF ;  the  whole 
weight  of  the  body,  therefore,  presses  upon  the  plane  pi*.  Let 
GE  represent  this  weight,  which  resolve  into  two,  GF,  FE  ;  GF 
will  represent  the  reaction  of  the  plane  upon  the  body,  and  FE 
will  represent  a  force  which  tends  to  make  the  body  slide  down 
the  plane.  Hence  it  appears,  that  the  body  will  have  a  ten- 
dency either  to  roll  or  slide,  according  as  the  angle  of  the  plane's 
inclination  is  greater  or  less  than  45°. 

In  considering  the  circumstances  under  which  a  body  would 
slide  or  roll  down  an  inclined  plane,  it  should  be  observed,  that 
if  the  surfaces  of  the  body  and  the  plane  be  perfectly  smooth,  no 
rolling  will  take  place,  whatever  be  the  angle  of  inclination  of 
the  plane.  To  give  a  body  a  tendency  to  rotary  motion  about 
its  center  of  gravity  (G),  it  is  evident  that  there  must  be  some 
mutual  action  between  the  surface  of  the  body  and  the  surface 
of  the  plane,  (such  as  that,  for  instance,  which  arises  from  friction, 
or  the  unrolling  of  a  rope ;)  if  there  be  not  some  such  action  as 
this,  all  the  parts  of  the  body  being  equally  accelerated,  the 
body  will,  under  all  circumstances,  slide  down  the  plane. 

76.  QUESTIONS  ON  THE  CENTER  OF  GRAVITY. 

1.  If  three  equal  bodies  be  placed  at  the  angles  of  any  triangle ; 
show  that  the  common  center  of  gravity  of  those  bodies  is  in  the 
same  point  with  the  center  of  gravity  of  the  triangle. 

2.  Four  bodies  A,  B,  C,  D,  weighing  respectively  2,  3,  6,  and 
8  pounds,  are  placed  with  their  centers  of  gravity  in  a  right  line, 
at  the  distance  of  3,  5,  7,  and  9  feet  from  a  given  point :  What 
is  the  distance  of  their  common  center  of  gravity  from  that  given 
point ;  and  between  which  two  of  the  bodies  does  it  lie  ? 


72  NATURAL    PHILOSOPHY. 

Ans.  Between  C  and  D ;  and  its  distance  from  the  given  point 
t&feet. 

3.  The  bodies  A,  B,  C,  weighing  respectively  5,  3,  and  12 
pounds,  are  so  placed,  that  AB=8  feet,  AC=4  feet,  and  the  angle 
BAG  is  a  right  angle :  What  is  the  distance  of  their  common 
center  of  gravity  from  the  body  C  ?  Ans.  2  feet. 

4.  Supposing  the  height  of  the  cylinder  in  Exam.  2,  (Art.  75.) 
to  be  only  twice  the  diameter  of  its  base  :  What  is  the  limit  of 
its  angle  of  inclination  before  it  is  in  danger  of  falling  ? 

Ans.  60°. 


EFFECT   PRODUCED    UPON    THE    COMMON  CENTER  OF   GRAVITY  OF  A  SYSTEM 
OF  BODIES,  WHEN  SOME  OR  ALL  OF  THEM  ARE  ACTUALLY  IN  MOTION. 

77.  If  two  bodies  approach  to  or  recede  from  each  other,  with 
velocities  inversely  proportional  to  their  weights,  their  common  center 
of  gravity  will  remain  at  rest. 

Fig.  52. 
a      A      a  b  B  b 

| — •-H 1 1 o 1 

G 

Let  A  and  B  (Fig.  52)  be  two  unequal  bodies ;  then  if  they 
approach  to  or  recede  from  each  other  with  velocities  inversely 
proportional  to  their  weights,  (in  which  case  their  momenta  will 
be  equal  by  Art.  14,)  their  common  center  of  gravity  G  will  re- 
main at  rest.  For  take  Aa  :  Eb  : :  B  :  A,  and  suppose  A  to  move 
through  Aa  while  B  moves  through  Eb,  then  (since  Vac  S  when 
T  is  given)  velocity  of  A  :  velocity  of  B  : :  Aa  :  Eb  : :  B  :  A,  (Art. 
67  ;)  hence  we  have 

AG  :  BG  : :  B  :  A,  and  Aa  :  Eb  : :  B  :  A  ; 
.-.  *  AG±Aa  :  BG±B6  : :  B  :  A,  i.  e.  aG  :  6G  : :  B  :  A ; 
from  which  it  appears  that  G  is  their  common  center  of  gravity 
when  the  bodies  are  arrived  at  a  and  b,  i.  e.  the  center  of  gravity 
has  remained  at  rest  while  the  bodies  have  approached  to  or  re- 
ceded from  each  other  through  the  spaces  Aa,  Bb. 

78.  When   one    body  moves  uniformly,   describing  any  figure 
around  another  at  rest,  the  center  of  gravity  of  the  two  bodies  de- 
scribes a  similar  figure  around  the  central  body. 

Let  A  (Fig.  53)  remain  at  rest,  while  B  moves  uniformly  along 
the  sides  BC,  CD,  DE,  of  the  polygon  BCDE.  When  the  body 
arrives  at  C,  join  AC,  and  take 

AK  :  KG  : :  B  :  A,  or  AK  :  AC  : :  B  :  A+B, 

then  K  will  be  the  place  of  the  center  of  gravity,  (Art.  68.) 
When  the  body  arrives  at  D,  E,  join  also  AD,  AE,  and  divide 
them  in  the  points  L,  M,  in  the  ratio  of  B  :  A,  then  will  L,  M  be 

»  Algebra,  388. 


MECHANICS.  73 


the  position  of  the  center  of  gravity  at  the  end  of  those  respec- 
tive times.     Let  GK,  KL,  LM  be  now  joined ;  and  since 
AG  :  AB  : :  B  :  A+B,  and  AK  :  AC  : :  B  :  A+B, 

.-.  AK  :  AC  : :  AG  :  AB ; 

hence  GK  is  parallel  to  BC,  and  the  triangle  AGK  similar  to  the 
triangle  ABC.  In  the  same  manner  it  may  be  proved  that  the 
triangles  AKL,  ALM  are  respectively  similar  to  ACD,  ADE, 
and  the  whole  figure  AGKLM  to  the  polygon  ABCDE.  While 
the  body  B  therefore  moves  uniformly  along  the  sides  of  the 
polygon  BCDE,  the  common  center  of  gravity  G  describes  with 
a  uniform  motion  a  similar  polygon  GKLM  ;  and  since,  from  the 
nature  of  similar  figures,* 

GK+KL+LM  :  BC+CD+DE  : :  (AG  :  AB.  i.  e.)  B  :  A+B, 
the  velocity  of  the  center  of  gravity  will  be  to  the  velocity  of 
the  body  B  as  B  to  A+B.  (Art.  12.)  Suppose  now  the  number 
of  the  sides  of  the  polygon  BCDE  to  be  increased  without  limit, 
so  that  it  may  be  considered  as  assuming  the  form  of  a  curve, 
then  shall  we  come  to  this  general  conclusion,  that,  while  the 
body  B  proceeds  uniformly  along  the  perimeter  of  the  figure 
BCDE,  whether  rectilinear  or  curvilinear,  the  center  of  gravity 
G  will  describe  with  a  uniform  motion  a  similar  figure  GKLM, 
with  a  velocity  which  is  to  that  of  B,  as  B  is  to  A+B. 

79.  When  a  system  of  bodies  are  in  motion,  their  common,  center 
of  gravity  will  move  in  the  same  manner  as  if  a  body  equal  to  the 
sum  of  the  bodies  were  placed  in  that  point,  and  the  same  motions 
were  communicated  to  it  as  are  communicated  to  the  bodies  separately. 

Let  us  take  the  case  of  three  bodies.  A,  B,  C,  moving  with  uni- 
form velocities,  in  equal  successive  parts  of  time,  through  the 
spaces  Aa,  Eb,  Cc.  Let  G  be  the  position  of  the  common  center 
of  gravity  of  the  three  bodies,  and  g  that  of  B  and  C,  before  they 
begin  to  move  ;  then  (Art.  68)  Gg  :  Ag : :  A  :  A+B+C.  While 
A  moves  from  A  to  a,  B+C  may  be  considered  as  at  rest  in  g, 
' .  (Art.  78,)  the  common  center  of  gravity  (G)  will  in  the  same 

*  Algebra,  388. 

10 


74  NATURAL    PHILOSOPHY. 

time  describe  GK  parallel  to  Aa,  and 

GK  :  A<z : :  Gg  :  Ag : :  A  :  A+B+C. 

When  A  is  arrived  at  a,  join  aC  and  BK. ;  produce  BK  till  it 
meets  aC  in  m,  then  m  will  be  the  center  of  gravity  of  A  and  C  ;* 
join  mb,  then  while  B  moves  from  B  to  b,  the  common  center  of 
gravity  will  describe  KL  parallel  to  B6,  and 

KL  :  Eb : :  mK  :  mB  : :  B  :  A+B+C. 

When  B  is  arrived  at  b,  join  ab,  CL  ;  produce  CL  till  it  meets 
ab  in  n,  then  n  will  be  the  center  of  gravity  of  A  and  B  ;  join  nc, 
then  while  C  moves  from  C  to  c,  the  common  center  of  gravity 
will  describe  LM  parallel  to  Cc,  and 

LM  :  Cc : :  nL  :  nC : :  C  :  A+B+C. 
Fig  54. 


While  the  bodies  A,  B,  C,  therefore,  in  equal  successive  parts  of 
time,  move  uniformly  through  the  spaces  Aa,  B&,  Cc,  their  com- 
mon center  of  gravity  will  in  the  same  time  describe  the  polygon 
GKLM,  whose  sides  GK,  KL,  LM,  are  respectively  parallel  to 
Aa,  B&,  Cc,  and  bear  to  them  the  ratio  of  A,  B,  and  C  to  A+B+C. 

80.  If,  instead  of  moving  in  successive  intervals  of  time,  the 
three  bodies  A,  B,  C,  were  all  to  begin  to  move  at  the  same  in- 
stant, and  describe  the  lines  Aa,  B6,  Cc,  cotemporaneously  ;  let 
us  then  consider  what  effect  would  be  produced  upon  their 
common  center  of  gravity.  Now  since  GK,  Aa,  are  described 
in  the  same  time,  calling  the  velocity  of  the  common  center  of 

*  For,  when  A  moves  to  a,  the  center  of  gravity  of  A  and  C  is  somewhere  in  the 
line  Ca.  But  when  A  moves  to  a,  the  center  of  gravity  of  the  three  bodies  moves  to 
K  ;  therefore  the  center  of  gravity  of  A  and  C  must  also  be  in  the  line  BK  produced, 
since  it  must  be  such  a  point  that  A  and  C  when  placed  there  shall  balance  B.  It 
must  therefore  be  hi  the  intersection  of  Ca  and  BK,  or  at  m. 


MECHANICS.  75 

gravity  v,  and  that  of  the  body  A,  V,  then  v  :  V  : :  GK  :  Aa  : :  A 
:  A+B+C  ;  hence  A  x  V=(A+B+C)  xv  ;  i.  e.  the  momentum  of  A 
(Art.  14)  is  equal  to  the  momentum  of  a  body  equal  to  A+B+C 
moving  with  the  velocity  v ;  the  same  force  which  causes  the  body 
A  to  move  over  Aa,  would  in  the  same  time  cause  a  body  equal  to 
A+B+C  to  move  over  GK.  For  the  same  reason,  the  forces  which 
impel  B  and  C  over  Eb,  Cc,  are  such  as  would  in  the  same  time 
cause  a  body=A+B+C  to  move  over  KL,  LM.  Hence  it  appears 
that  the  motion  of  the  common  center  of  gravity  along  the  sides  of 
the  polygon  GKLM,  is  analogous  to  the  motion  of  a  body  equal 
to  A+B+C  acted  upon  by  three  forces  which  would  carry  it  over 
GK,  KL,  LM  in  the  same  time  that  they  would  carry  the  bodies 
A,  B,  C,  over  the  spaces  Aa,  Eb,  Cc,  respectively.  But  a  body 
acted  upon  at  once  by  these  forces  would  (Art.  45)  describe  the 
other  side  GM  of  the  polygon  GKLM  in  the  same  time  that  it 
would  describe  either  of  the  sides  GK,  KL,  LM,  when  the  forces 
act  separately  ;  if  the  bodies  A,  B,  C,  therefore,  move  cotempo- 
raneously,  their  common  center  of  gravity  will  describe  the  line 
GM,  while  the  bodies  themselves  describe  the  three  lines  Aa,  Eb, 
Cc,*  and  the  same  reasoning  is  applicable  to  any  number  of 
bodies. 

81.  Hence,  in  the  first  place,  if  the  bodies  which  compose  a 
system  move  uniformly  in  right  lines,  then  their  common  center 
of  gravity  will  either  remain  at  rest,  or  will  move  uniformly  in  a 
right  line ;  for  if  a  body  equal  to  the  sum  of  the  bodies  were 
placed  in  that  center,  and  then  acted  upon  by  the  same  forces 
which  cause  the  bodies  to  move  separately  in  right  lines,  it 
would  either  remain  at  rest,  (viz.  when  the  forces  counteract 
each  other,)  or  would  describe  uniformly  the  remaining  side  of  a 
polygon,  whose  other  sides  represent  the  quantity  and  direction 
of  the  several  forces  acting  upon  it.  In  the  second  place,  the 
common  center  of  gravity  of  the  system  will  not  be  affected  by 
the  mutual  action  of  the  bodies  upon  each  other  ;  for  action  and 
reaction  being  equal,  the  effect  produced  upon  the  common  cen- 
ter of  gravity  by  such  mutual  action,  will  only  be  that  of  two 
equal  and  opposite  forces  acting  upon  a  body  equal  to  the  sum 
of  the  bodies  placed  in  that  center  ;  which  would  evidently  not 
disturb  its  state,  either  of  motion  or  quiescence.  Lastly,  if  the 
motion  of  the  bodies  in  these  right  lines  were  to  cease,  and  they 
were  left  to  the  mutual  attraction  of  each  other,  then  their  com- 
mon center  of  gravity  would  remain  at  rest,  and  the  bodies 
would  approach  each  other,  in  lines  drawn  to  it  from  their  re- 

*  We  have  here  supposed  the  bodies  A,  B,  C,  to  have  their  centers  of  gravity  in  the 
same  plane  ;  in  which  case  it  is  evident  that  the  motion  of  their  common  center  of 
gravity  will  be  in  the  same  plane.  If  the  motion  of  the  bodies  be  in  different  planes, 
then  the  value  of  each  line  GK,  KL,  LM,  might  be  found  as  before  ;  but  as  they  will 
then  lie  in  different  planes,  the  resulting  quantity  GM  must  be  ascertained  according 
to  the  principles  laid  down  in  Arts.  43  and  44. 


76 


NATURAL   PHILOSOPHY. 


spective  centers  of  gravity,  and  all  collect  together  in  that  com- 
mon center. 

82.  EXAMPLES. 

1.  Let  two  equal  bodies  A  and  B,  move  from  the  point  D,  with 
the  same  uniform  velocity,  along  the  sides  DE,  DF,  of  the  iso- 
sceles triangle  DEF,  whose  angle  EDF=120°  :  It  is  required  to 
compare  the  velocity  of  their  common  center  of  gravity  with 
that  of  either  of  the  bodies  A  or  B. 


. 

K  F 

When  the  bodies  are  arrived  at  the  points  A,  B,  (Fig.  55,)  join 
AB  ;  and  since  the  bodies  move  with  the  same  uniform  velocity, 
DA=DB  ;  .-.  DA  :  DB  : :  DE  :  DF,  and  AB  is  parallel  to  EF. 
Again,  because  A=B,  the  center  of  gravity  G  will  bisect  AB ; 
hence,  while  the  bodies  move  uniformly  along  DE,  DF,  the  cen- 
ter of  gravity  will  move  through  the  line  DK,  which  bisects  AB, 
EF  at  right  angles.  Now  EDF=120°,  .-.EDK=60°,  and  DEK 
=30°  ;  but  since  DE,  DK  are  described  in  the  same  time,  velo- 
city of  A  :  velocity  of  G  : :  DE  :  DK  : :  rad.  :  sin.  30°  : :  2  :  1 ;  the 
center  of  gravity,  therefore,  moves  with  half  the  velocity  of  either 
of  the  bodies  A  and  B. 

2.  Let  the  two  bodies  A  and  B  be  placed  at  the  extremity  A 
of  the  diameter  of  the  circle  ADF,  and  then  let  B  describe  the 
circle  ADF  while  A  remains  at  rest  in  the  point  A :  In  what 
manner  will  their  common  center  of  gravity  move  ? 

Let  B,  b,  (Fig.  56,)  be  any 
two  positions  of  the  body  B, 
and  G,  g,  the  corresponding 
positions  of  the  common  cen- 
ter of  gravity  of  A  and  B. 
Join  B6,  Gg ;  then  by  Art. 
78,  Gg  is  parallel  to  B6,  /. 
AG,g-=AB6,  hence  AD&,  Adg, 
are  similar  segments  of  cir-A 
cles ;  and  when  B  has  de- 
scribed the  semicircle  ADE, 
the  center  of  gravity  will 
have  described  the  semicir- 
cle Ade  ;  and  so  for  the  semi- 
circles on  the  other  side  of 
AE.  Now  Ae:eE::B:A, 


MECHANICS. 


77 


/.  Ae  :  AE  : :  B  :  A+B  ;  hence  while  B  describes  the  circle  ADP, 
the  common  center  of  gravity  of  A  and  B  will  describe  the 
circle  A.df,  whose  diameter  :  diameter  of  ADF  :  :  B  :  A+B. 
This  conclusion,  indeed,  follows  immediately  from  the  reasoning  in 
Art.  78,  for  it  was  there  shown  that  the  whole  figure  described 
by  the  common  center  of  gravity,  is  similar  to  that  which  the 
moving  body  B  describes. 

3.  Two  bodies  A  and  B,  begin  to  move  in  opposite  directions 
at  the  same  instant  from  the  extremity  D  of  the  diameter  DE 
of  the  circle  DAEB,  and  continue  to  move  on  with  the  same 
uniform  velocity  till  they  meet  in  E  ;  they  pass  each  other  at  E, 
and  then  continue  to  move  on  till  they  arrive  at  the  point  D, 
whence  they  set  off :  What  is  the  course  of  the  common  center 
of  gravity  during  this  revolution  of  the  two  bodies  ? 


Fig.  57. 


Suppose  the  bodies  arrived  at 
the  position  AB,  (Fig.  57,)  then 
since  DA=DB,  the  line  AB  will 
be  bisected  by  DE  in  N.  Let 
G  be  the  common  center  of 
gravity  of  A  and  B,  then 

A+B  :  B  : :  AB  :  AG,  .-. 
i(A+B)  :  B  : :  iAB  (or  AN)  :  AG  ;  D 
hence  AN  :  AG  : :  A+B  :  2B,  and 
.-.  AN  :  AN- AG  (GN)  : :  A+B  : 
A+B-2B  (A-B);  i.  e.  AN: 
GN  in  the  given  ratio  of  A+B 
:  A  — B  ;  consequently  while  the 
bodies  A  and  B  describe  respect- 
ively the  semicircles  DAE,  DBE,  their  common  center  of  gravity 
describes  the  semi-ellipse  DGKE.  In  the  same  manner  it  may 
be  proved  that  while  A  and  B  describe  the  semicircles  EBD 
EAD,  their  common  center  of  gravity  would  describe  a  semi- 
ellipse  ELMD,  equal  and  similar  to  DGKE.  While  the  bodies  A 
and  B  therefore  perform  their  respective  revolutions,  their  com- 
mon center  of  gravity  will  describe  the  ellipse  DGEM,  whose 
major  axis  :  minor  axis  : :  (AN  :  GN  : :  )  A+B  :  A  — B. 

Cor.  If  the  bodies  be  equal,  A  —  B=0,  and  the  ellipse  becomes 
a  straight  line.  Indeed  it  is  evident  that  in  this  case  the  common 
center  of  gravity  would  move  in  a  line  which  always  bisects  AB, 
ab,  i.  e.  in  the  diameter  DE. 

4.  Three  bodies  A,  B,  C,  at  the  same  instant  begin  to  move 
uniformly  from  the  three  angles  of  a  given  triangle,  and  in  the 
same  time  change  places  in  the  direction  ABC  :  How  will  their 
common  center  o£  gravity  be  affected  by  this  motion  of  the  bodies  ? 


78  NATURAL   PHILOSOPHY. 

Let  G    (Fig.    58,)   be    their  , 

common  center  of  gravity,  and  •»  .1g.  58. 

suppose  the  bodies  first  to  move 
in  succession.  While  A  moves 
from  A  to  B,  their  common 
center  of  gravity  will  (by  Art. 
78)  describe  GK  parallel  to 
AB,  and  GK  :  AB  :  :  A  :  A+ 

B  moves  from  B  to  C,  the  center  of  gravity  will  describe 

T>     y.    Dp 

KL—  ;  and  if  L  M  be  drawn  parallel  to  CA,  and  equal 


to  i  nip*  while  C  moves  from  C  to  A,  the  center  of  gravity 

will  describe  LM.  Suppose  now  the  bodies  to  move  cotempo- 
raneously,  then  their  common  center  of  gravity  will  describe 
GM,  (the  remaining  side  of  the  polygon  GKLM,)  while  the  bo- 
dies change  places  in  the  direction  ABC.  (Art.  44.) 

To  find  the  actual  value  of  GM,  we  have  KL,  LM,  and  the 
angle  KLM  (—  ACB)  given,  from  which  MK  and  MKL  may  be 
found  ;  but  GKM=GKL  (or  ABC)-MKL  ;  in  the  triangle  GKM 
there  are  therefore  given  GK,  KM,  and  GKM,  from  which  GM 
may  be  determined. 

Cor.  If  A=B=C,  then  GK=£AB,  KL=iBC,  and  LM=iAC, 
.-.  GK,  KL,  LM,  are  to  each  other  as  AB,  BC,  AC  ;  and  since  the 
angles  GKL,  KLM  are  respectively  equal  to  ABC,  BCA,  the 
three  lines  GK,  KL,  LM  will  form  a  triangle  similar  to  the  tri- 
angle ABC.  GM  therefore  in  this  case  is  equal  to  0,  and  the 
body  remains  at  rest.  This  also  follows  from  the  gem  ral  theo- 
rem in  Art.  79  ;  for  the  common  centre  of  gravity,  being  under 
the  same  circumstances  as  a  body  acted  upon  by  three  forces 
which  are  to  each  other  as  the  three  sides  of  a  triangle  taken 
in  order,  will,  by  Art.  55,  remain  at  rest. 

83.  The  distance  of  the  common  center  of  gravity  of  any  numbei 
of  bodies  or  particles  of  matter  from  a  plane  given  in  position,  is 
equal  to  the  sum  of  the  products  arising  from  multiplying  each 
body  into  its  distance  from  the  plane,  divided  by  the  sum  of  the 
bodies. 

Let  p,  p',  p",  (Fig.  59,)  be  any  number  of  small  bodies  or 
particles  of  matter,  and  ABCD  a  plane  placed  in  any  position 
with  respect  to  them.  Join  pp',  and  let  g  be  the  common  center 
of  gravity  of  p  and  p'  ;  draw  px,  gJc,  p'x'  at  right  angles  to  the 
plane  ABCD,  and  consequently  parallel  to  each  other  ;  join  xx't 
and  since  the  points  p,  g,  p'  are  in  a  straight  line,  the  points  x,  k, 
x'  will  also  be  in  a  straight  line,  and  therefore  xx  will  pass 


MECHANICS. 


through  k.  Join  gp",  and  let  G 
be  the  common  center  of  grav- 
ity of  p,  p',  p"  ;  draw  GK,  p'x'-1 
perpendicular  to  the  plane  ;  and 
through  g  draw  mn  parallel  to 
xx1  meeting  px  produced  in  n. 

Now  p:p'  ::  p'g:  pg  ::  (by 
sim.  triangles)  p'm  :  pn  ; 
.'.p  xpn=p'  xp'm, 
or  px  (nx—px)  =p'  x  (p'x'  —  mx')  ; 
but  nx=gk—7nx', 

.'.px(gk—px)=p'x(p'x'-gkj} 
and  (p+p')xgk=pxpx-\-p'xp'x 


Fig.  59. 


'  m   \ 


.  ~f- 

for  the  same  reason, 

ifp-\-p'  is  placed  at  g,  we  have 

(p+p1)  xgk+p"  xp"x 


_p  xpx+p'  xp'x'+p"  Xp"x" 

p+p'+p" 
and  thus  we  might  proceed,  whatever  be  the  number  of  particles. 


CHAPTER  V. 

OF  THE  COLLISION  OF  BODIES. 

84.  BODIES  are  divided  into  elastic  and  inelastic.  Elastic  bod- 
ies are  such  as,  when  compressed,  restore  themselves  to  their  for- 
mer state.  Inelastic  bodies  are  such  as  do  not  thus  restore  them- 
selves. Thus,  sponge,  wool,  cotton,  and  India  rubber,  are  more 
or  less  elastic  ;  and  air,  which  restores  itself  with  a  force  equal 
to  that  which  compresses  it,  is  perfectly  elastic.  But  lead  and 
clay  are  inelastic  bodies,  since,  when  they  impinge  upon  one 
another,  they  do  not  rebound.  Ivory,  glass,  and  steel,  are  among 
the  most  elastic  substances  known.  If  we  suspend  two  ivory 
balls  by  strings  of  the  same  length,  and  let  them  fall  upon  one 
another,  (as  in  Fig.  4,  page  29,)  we  may  render  the  compression 
which  they  undergo  on  meeting  apparent,  by  dotting  the  points 
of  contact  with  ink  ;  after  impact,  these  dots  will  be  enlarged  in 
a  circular  space  around  the  original  point.  Experiments  on  this 
subject  are  supposed  to  be  made  with  two  spheres  or  balls  of  the 
same  density,  moving  uniformly  in  the  line  which  joins  their 
centers  of  gravity. 


80  NATURAL   PHILOSOPHY. 

85.  When  one  INELASTIC  body  strikes  upon  c-nother  at  rest,  or 
moving  with  less  velocity  in  the  same  direction,  the  two  bodies  move 
on  together  as  one  mass,  with  a  velocity  equal  to  the  sum  of  the  mo 
menta  divided  by  the  sum  of  the  bodies. 

Thus,  let  A,  B  represent  the  two  bodies,  and  a,  b,  their  respect- 
ive velocities  ;  then  Aa  will  be  the  momentum  of  A,  and  Eb  that 
of  B.  (Art.  14.)  The  sum  of  their  momenta  is  Aa-{-Eb.  Let  v 
be  the  common  velocity  after  impact  ;  then  (A+B)xv=the  mo- 

mentum of  the  mass.     Then  Aa+B6=(A+B)  xv  ;  .'.v= 


If  B  is  at  rest,  then  the  common  velocity  equals  the  momentun 
of  A  divided  by  the  sum  of  the  bodies  ;  for  then  Eb  becomes  0, 


The  velocity  lost  by  A  equals  the  product  of  B  into  the  DIFFER- 
ENCE of  their  velocities,  divided  by  the  sum  of  the  bodies  ;  and 
that  gained  by  B,  equals  the  product  of  A  into  the  DIFFERENCE  of 
the  velocities  divided  by  the  sum  of  the  bodies.  For,  the  veloci- 

Aa+B6    E(a—b) 
ty  lost  by  A=a—  v=  a  --       „  =    IIP*     The  velocity  gained 

.    Aa+Eb     .     A(a-b) 


When  B  is  at  rest,  these  expressions  become  .  .  p  and  >     ~ 

86.  When  the  bodies  move  in  OPPOSITE  directions,  the  common  ve- 
locity after  impact  equals  the  difference  of  their  momenta  divided 
by  the  sum  of  the  bodies. 

The  momentum  of  the  mass  after  impact  is  the  difference  of 
their  momenta  before  impact.  Hence  Aa—  Eb=(A-\-E)  xv,  .• 

u—   A  IP  '    The  velocity  lost  by  A  equals  the  product  of  B  into 

the  SUM  of  the  velocities  divided  by  the  sum  of  the  bodies  ;  and 
that  gained  by  B  equals  the  product  of  A  into  the  SUM  of  the  ve- 
locities, divided  by  the  sum  of  the  bodies. 

Aa-Eb    E(a+b) 
For,  the  velocity  lost  by  A=a—v=a  --       „  =   ^        . 

The  velocity  gained  byB  (in  the  direction  of  A)=  —  \  -^. 

A~ri> 

When  the  two  bodies  are  equal,  and  meet  with  equal  velocities, 
the  expression  v=  ^~  becomes  v=0,  and  both  bodies  remain 


at  rest.  Since,  in  this  case,  Aa=Eb  .:  A  :  E  ::&:«;  therefore, 
conversely,  when  bodies  move  before  impact  with  velocities  inversely 
proportional  to  their  quantities  of  matter,  they  will  be  at  rest  after 


MECHANICS.  81 

impact.  The  same  conclusion  may  be  drawn  from  the  consider- 
ation that,  in  this  case,  the  bodies  would  meet  with  equal  mo- 
menta. (See  Art.  14.) 

87.  EXAMPLES  FOR  INELASTIC  BODIES. 

1.  A,  weighing  3  oz.,  and  moving  10  feet  per  second,  overtakes 
B,  weighing  2  oz.,  and  moving  3  feet  per  second :  What  is  the 
common  velocity  after  impact  ?  Ans.  7±  feet  per  second. 

2.  A  weight  of  7  oz.,  moving  1 1  feet  per  second,  strikes  upon 
another  at  rest  weighing  15  oz. :  Required  the  velocity  after  im- 
pact? Ans.  3%  feet  per  second. 

3.  A  weighs  4  and  B  2  pounds ;  they  meet  in  opposite  direc- 
tions, A  with  a  velocity  of  9,  and  B  with  one  of  5  feet  per  second : 
What  is  the  common  velocity  after  impact  ? 

Ans.  4|  feet  per  second. 

4.  A =7  pounds,  B=4  pounds ;  they  move  in  the  same  direction, 
with  velocities  of  9  and  2  feet  per  second :  Required  the  velocity 
lost  by  A  and  gained  by  B  ?  Ans.  A  2T6T,  B  4ysT. 

5.  A  body  moving  7  feet  per  second,  meets  another  moving  3 
feet  per  second,  and  thus  loses  half  its  momentum :  What  are 
the  relative  magnitudes  of  the  two  bodies  ? 

Ans.  A  :  B  : :  13  :  7. 

6.  A  weighs  6  pounds  and  B  5  ;  Bis  moving  7  feet  per  second, 
in  the  same  direction  as  A  ;  by  collision  B's  velocity  is  doubled  : 
What  was  A's  velocity  before  impact  ? 

Ans.  19|  feet  per  second. 

88.  In  the  collision  of  ELASTIC  bodies,  the  velocity  lost  by  the  one 
and  gained  by  the  other,  is  twice  that  which  it  would  have  been,  had 
the  bodies  been  inelastic. 

According  to  the  definition  of  elasticity,  the  body  restores  itself 
with  a  force  equal  to  that  which  compresses  it ;  consequently  as 
much  momentum  is  exerted  in  the  restitution  as  in  the  compres- 
sion. In  a  given  body,  therefore,  the  velocity  of  restitution  is 
equal  to  that  of  compression.  Suppose,  for  example,  that  a  ball 
of  lead  A,  strikes  upon  another  B  ;  then  what  B  gains  A  loses  by 
reaction,  and  both  bodies  move  on  together ;  but  when  a  ball  of 
ivory  (supposed  perfectly  elastic)  impinges  on  another,  it  not 
only  loses  the  momentum  which  it  at  first  imparted  to  B,  but  the 
latter,  in  restoring  itself  after  compression,  exerts  a  force  equal 
to  that  of  reaction,  and  therefore  destroys  as  much  more  of  the 
motion  of  A.  Again  A,  while  receiving  this  second  impulse  from 
B,  reacts  with  an  equal  force,  and  thus  doubles  the  effect  of  its 
impulse  upon  B. 

Distinguishing  the  corresponding  elastic  body  by  an-  accent, 
since,  the  direction  being  the  same,  the  velocity  lost  by  A= 

11 


82  NATURAL   PHILOSOPHY. 

B^~5);  .\  that  lost  by  A>=2B(^~6)  ...  the  velocity  of  A'  after 

A+Jo 


2B(a-6)    (A-B)«+2B& 
impacts  > 


So  the  velocity  of  B' 


p. 
A-pB 

When  the  directions  are  opposite, 
_,,         .     .^    .    ^,      .     B(a+6)     „       ,    2B(«+&)     __ 
The  velocity  lost  by  A=  .v  .T.  y.    By  A'=  —  rr^.    Hence, 
A+r>  A-plS 

*-';.;      ,  A,    ft     •  2B(«+6)    (A-B)<z-2B& 

velocity  of  A'  after  impact  =a  --  rr^        -  A  To  -  ' 

A  -pi)  A-p-D 

And  velocity  of  B'  after  impact=(A""f)|6+2Aa. 

A-pB 

89.  When  equal  elastic  bodies  impinge  upon  one  another,  each 
moves  after  impact  with  the  previous  velocity  of  the  other  body. 

For  if  B=A,  then  A—  B  or  B—  A  are  each  equal  to  0  ;  .•.  when 
the  bodies  move  in  the  same  direction  before  impact,  the  velocity 

QT}7  ORA 

of  A  after  impact=          ~7m'~^  >  an^  the  velocity  of  B  after 
~~ 


2Aa  Tf.  xl  ,    „ 

impact—          =-—  —=a.     If  they  move  before  impact  in  oppo- 
A-rr>      2A 

site  directions,  then  the  velocity  of  A  after  impact— 

—  2B&         ,        ,,,         ,     ..      f-n    f      .  2A« 

—  -—  =  —o.  and  the  velocity  of  B  after  impact^ 


.  -      . 

2B  A+B      2A 

Hence,  in  all  cases  when  the  bodies  are  equal,  they  move  after 
impact  with  interchanged  velocities  ;  that  is,  when  the  direc- 
tions are  the  same,  A  moves  on,  after  impact,  with  the  velocity 
of  B,  and  B  moves  on  with  the  velocity  of  A  ;  and  when  the 
directions  are  opposite,  A  returns  with  the  velocity  of  B,  and  B 
returns  with  the  original  velocity  of  A. 

90.  When  equal  elastic  bodies,  moving  with  equal  velocities,  in 
opposite  directions,  meet,  each  is  rejected  back  with  its  original 
velocity. 

For,  by  Art.  89,  the  velocity  of  A  after  impact  =  —  b,  and  that 
of  B=a  ;  and'since  a=b,  each  returns  with  its  previous  velocity. 

If  B  rests  before  impact,  then  6=0  ;  .*.  the  Vy  of  A  after  impact 

—     ~       ,  and  velocity  of  B   after  impact=—  -^.     If  A  be 
- 


greater  than  B,  then-      ••       •  is  positive,  .'.  A  moves  after  impact 
A"T~B 

in  the  same  direction  as  it  did  before  with  Vy=-  .  .  g    >  and  B 


MECHANICS.  83 

precedes  it  with  a  velocity^^^  which  is  greater  than  a.)  If  A 

be  less  than  B,  their   .""?;    is  negative,  .'.  A  is  reflected  back  by 
ATJJ 

its  impact  upon  B  with  a  velocity—  '  a  ;  and  B  moves  for- 

A~ro 

ward  in  A's  original  direction  with  a  velocity  =•          (which  is 

AT~JJ 

less  than  a.) 

91.  If  one  elastic  body  strikes  on  another  equal  to  it  at  rest,  the 
Jirst  body  will  be  brought  to  a  state  of  rest,  while  the  second  will 
move  on  with  the  velocity  of  the  first. 

If  A  be  equal  to  B,  then(A~^)g  (which  is  the  velocity  of  A 


after  impact)  =0,  and  —  —  (which  is  the  velocity  of  B  after  im- 

pact)=      .    =a;  i.  e.  if  B  rests  before  impact,  then  A  will  rest 
£  A. 

after  impact  ;  and  B  will  move  forward  in  A's  direction  with  A's 
velocity  before  impact. 

Let  there  be  a  row  of  equal  elastic  balls  A,  B,  C,  &c.  .  .  X, 
(Fig.  60,)  placed  contiguous  to  each  other;  then  (by  Art.  89,)  if 
A  is  moved  from  its  position  and  made  to  impinge  upon  B,  it  will 
rest  after  impact,  and  B  will  have  a  tendency  to  move  on  with 
A's  velocity  ;  after  the  impact  of  B  upon  C,  it  will  remain  at 
rest,  and  C  have  a  tendency  to  move  on  with  A's  velocity  ;  after 
the  impact  of  C  upon  D,  it  will  remain  at  rest,  and  D  will  have 
a  tendency  to  move  on  with  the  same  velocity  ;  and  so  the  mo- 
tion will  be  propagated  through  the  whole  row,  and  the  last  body 
X  will  move  forward  with  the  velocity  of  A,  all  the  others  re- 
maining at  rest. 

A  B      C      D     E  X 

o  ooooo  o  «.<* 

QQ     O     O  O        Fig.6l; 


° 

If  the  bodies  decrease  in  magnitude,  (Fig.  61,)  then,  since  A 
is  greater  than  B,  (by  Art.  90,)  the  velocity  communicated  to  B 
will  be  greater  than  that  of  A  ;  and  the  velocity  communicated 


84 


NATURAL   PHILOSOPHY 


from  B  to  C  greater  than  that  of  B,  &c. ;  so  that  the  last  body 
will  move  forward  in  the  direction  of  A's  motion  with  a  velocity 
much  greater  than  that  of  A,  and  the  other  bodies  will  follow  it 
in  such  a  manner,  that  the  velocity  of  each  succeeding  body 
shall  be  greater  than  that  of  the  preceding.  On  the  contrary, 
if  the  bodies  increase  in  magnitude,  (Fig.  62,)  since  A  is  less 
than  B,  (by  Art.  90,)  the  velocity  communicated  to  B  will  be  less 
than  that  of  A,  and  A  will  be  reflected  back  by  B  ;  for  the  same 
reason  the  velocity  communicated  from  B  to  C  will  be  less  than 
that  of  B,  and  B  will  be  reflected  back  by  C  ;  so  that  in  this  case 
all  the  bodies  will  move  backward  except  the  last,  and  that  will 
move  forward  in  the  direction  of  A's  original  motion,  but  with  a 
velocity  much  less  than  that  of  A. 

92.   When  a  row  of  bodies  are  in  geometrical  progression,  and 

'    the  first  impinges  on  the  second,  and  motion   is  thus  propagated 

through  the  series,  the  velocity  of  the  first  is  to  the  velocity  of  the 

(2 
1+ 

Let  the  series  be  A,  Ar,  Ar3.  .  .  .  Ar*"1. 
By  Art.  90,  when  A  impinges  on  B  at  rest  the  velocity  commu- 


.     -.•«..%,!    2Aa       2Aa 

mcated  to  B  is- — =-=- 


A+B    A+Ar    1+r 


2Ar 


Again,  the  velocity  imparted  to  C  is  PTT»"~A  2XTTT~ 

>a"  2a      22a 

-,&c. 


Hence  the  successive  velocities  are  a, 

from  which  it  appears  that  any  term  in  the  series  is  found  by 
multiplying  the  original  velocity  by  2,  raised  to  a  power  one  less 
than  the  number  of  terms,  and  divided  by  1+r  raised  to  the  same 

On- 1-2 

power.     Consequently,  the  last  term  is  -rr-t — ^— j.     Hence, 


V,  of  the  first:  V,  of  the  last  :  :  a  : 


93.  When  a  perfectly  elastic  body  impinges  on  a  perfectly 
smooth  plane,  it  makes  the  angle  of  reflexion  equal  to  the  angle  of 
incidence. 

A  Fig.  63.  a 


PC  L  a  N 

If  a  perfectly  elastic  body  impinges  perpendicularly  upon    •• 


MECHANICS.  85 

perfectly  smooth  plane,  then,  since  the  force  of  restitution  is 
equal  to  the  force  of  compression,  it  will  ascend  from  the  plane 
with  the  same  velocity  as  that  with  which  it  impinged  upon  it. 
But  if  moving  uniformly,  it  impinges  upon  the  plane  PN,  (Fig. 
63,)  in  the  oblique  direction  AL,  then  resolve  AL  into  two  AC, 
CL,  of  which,  as  in  the  former  instance,  the  perpendicular  part 
AC  will  not  be  destroyed,  but  will  represent  the  velocity  with 
which  the  body  ascends  from  the  plane  ;  and  CL  will  represent 
the  velocity  it  has  in  the  direction  of  the  plane,  the  same  as  be- 
fore. Take  therefore  Lc=LC,  and  from  c  draw  ca  at  right  angles 
to  Lc  and  equal  to  CA,  and  join  La  ;  then  La  will  represent  the 
direction  and  velocity  of  the  body  after  impact.  But  since  Lc, 
ca  are  equal  to  LC,  CA,  and  the  angles  Lea,  LCA  are  right 
angles,  La  will  be  equal  to  LA,  and  the  angle  aLc,  to  the  angle 
ALC  ;  hence  the  body  will  move  after  impact  with  the  same  ve- 
locity which  it  had  before  impact,  and  in  a  direction  making  the 
angle  of  reflexion  equal  to  the  angle  of  incidence. 

94.  EXAMPLES  FOR  ELASTIC  BODIES. 

1.  A  weighing  10  Ibs.  and  moving  8  feet  per  second,  impinges 
on  B  weighing  6  Ibs.  and  moving  in  the  same  direction,  5  feet 
per  second  :  What  are  the  velocities  of  A  and  B  after  impact  ? 

Ans.  A's=5£.     B's=8f  . 

2.  A  :  B  :  :  4  :  3  ;  directions  the  same  ;  velocities  5:4:  What 
is  the  ratio  of  their  velocities  after  impact  ?  Ans.  29  :  36. 

3.  A  weighing  4  Ibs.,  velocity  6,  meets  B  weighing  8  Ibs.,  ve- 
locity 4  :  Required  their  respective  directions  and  velocities  after 
collision  ? 

Ans.  A  is  reflected  back  with  a  velocity  of  7  ±,  and  B  with  a  velocity 


4.  A  and  B  move  in  opposite  directions  ;  A  equals  4B,  and 
b=2a  :  How  do  the  bodies  move  after  collision  ? 

Ans.  A  returns  with  i,  B  with  1  f  its  original  velocity. 

5.  There  are  ten  bodies  whose  magnitudes  increase  geometri- 
cally by  the  constant  ratio  3,  and  the  first  impinges  on  the  second 
with  the  velocity  of  5  feet  per  second  :  Required  the  motion  of 
the  last  body  1 

Ans.  The  last  body  would  move  with  the  velocity  of  Tf  ¥  feet  per 
second. 


86  NATURAL  PHILOSOPHY. 

CHAPTER  VI. 

OF  THE  LEVER. 

95.  IN  the  preceding  chapters,  the  motion  of  bodies  has  been 
supposed  to  arise  either  from  collision,  or  from  the  immediate 
action  of  one  or  more  forces.     We  now  proceed  to  consider  the 
effects  produced,  when  these  forces  are  made  to  act  by  the  inter- 
vention of  other  bodies.     These  intermediate  bodies  are  called 
Machines  ;  and  by  means  of  them  the  effect  of  a  given  force  may 
be  increased  or  diminished  in  any  given  ratio.     Machines  are 
divided  into  simple  and  compound. 

96.  The  simple  machines,  or  what  are  commonly  called  the 
MECHANICAL  POWERS,  are  six  in  number;  viz.  1.  The  Lever;  2 
The  Wheel  and  Axle  ;  3.  The  Pulley  ;  4.  The  Inclined  Plane  ;  5. 
The  Screw;    6.  The    Wedge.     In   philosophical   strictness,   the 
number  of  simple  machines  may  be  reduced  to  three ;  viz.  the 
lever,  the  inclined  plane,  and  the  cords  or  ropes  which  connect  the 
power  and  weight  with  the  different  parts  of  the  machine ;  for 
the  mechanism  of  the  wheel  and  axle,  and  of  the  pulley,  merely 
combines  the  principle  of  the  lever  with  the  tension  of  cords  ; 
the  properties  of  the  screw  depend  entirely  on  those  of  the  lever 
and  the  inclined  plane  ;  and  the  case  of  the  wedge,  so  far  as  it 
is  capable  of  mathematical  demonstration,  is  very  analogous  to 
that  of  a  body  sustained  between  two  inclined  planes.     Com- 
pound machines  are  formed  from  the  combination  of  two  or  more 
simple  ones.     But  it  is  not  the  object  of  this  treatise  to  enter 
upon  a  full  description  of  the  nature  and  use  of  compound  ma- 
chinery ;  our  intention  is  rather  to  explain,  upon  mathematical 
principles,  the  general  theory  of  mechanical  action. 

97.  The  LEVER  is  an  inflexible  bar  or  rod,  some  point  of  which 
being  supported,  the  rod  itself  is  movable  freely  about  that  point  as 
a  center  of  motion. 

This  center  of  motion  is  called  the  FULCRUM  or  PROP.  When 
two  forces  act  on  one  another  by  means  of  any  machine,  that 
which  gives  motion  is  called  the  POWER,  that  which  receives  it, 
the  WEIGHT. 

98.  In  treating  of  the  Mechanical  Powers,  the  first  inquiry  is, 
What  are  the  conditions  of  an  equilibrium  ?  that  is,  When  do  the 
power  and  weight  exactly  balance  each  other  ?     This  point  being 
ascertained,  any  addition  to  the  power  puts  the  weight  in  mo- 
tion.    The  investigation  first  proceeds  on  the  supposition  that 
the  action  of  the  mechanical  powers  is  not  impeded  by  their  own 


MECHANICS.  87 

weight,  or  by  friction  and  resistance,  a  suitable  allowance  being 
afterward  made  for  the  various  impediments. 

We  shall  begin  with  estimating  the  relation  between  the  forces 
acting  upon  the  arms  of  a  straight  lever,  which,  of  all  the  me- 
chanical powers,  is  the  most  simple. 

99.  If  any  two  forces,  acting  in  the  same  plane,  and  perpendicular 
to  the  extremities  of  a  straight  lever,  be  in  equilibrio,  they  will  be  to 
each  other  inversely  as  the  lengths  of  the  arms  upon  which  they 
respectively  act. 

Fig.  64. 
A  C  B 


1 


Let  ACB  (Fig.  64)  be  a  straight  lever,  supported  by  a  prop  or 
fulcrum  F,  and  movable  about  the  point  C  as  its  fulcrum.  From 
the  extremities  of  its  arms,  CA,  CB,  let  two  weights,  P,  W,  be 
suspended ;  and  suppose  them  to  be  in  equilibro  about  C,  the 
lever  itself  remaining  in  a  horizontal  position.  In  the  present 
instance,  let  us  also  suppose  that  the  lever  AB,  and  the  cords  AP, 
BW,  by  which  the  weights  are  suspended,  are  entirely  void  of 
gravity ;  in  which  case  it  is  evident  that  the  equilibrium  of  the 
bodies  does  not  at  all  depend  upon  the  length  of  the  cords  AP, 
BW ;  and  as  (Art.  70)  the  centers  of  gravity  of  the  bodies  P,  W, 
are  in  the  direction  of  the  lines  AP,  BW,  the  effect  will  be  the 
same  whether  the  bodies  are  suspended  by  the  strings  AP,  BW ; 
or  whether  they  are  placed  with  their  centers  of  gravity  in  the 
points  A,  B,  respectively.*  In  this  latter  case,  the  point  C  be- 
comes the  center  of  gravity  of  the  weights  P,  W,  and  conse- 
quently P :  W  : :  BC  :  AC,  (Art.  67.) 

But  it  is  evidently  quite  immaterial  to  the  truth  of  the  fore- 
going demonstration,  whether  the  equilibrium  of  the  lever  is  pro- 
duced by  the  force  of  gravity  of  the  two  weights,  P,  W,  or  by  the 
action  of  any  other  forces  in  the  directions  AP,  BW. 

100.  The  effect  of  any  forces  to  turn  the  lever  about  the  center  of 
motion,  is  measured  by  the  product  arising  from  multiplying  each 
force  into  the  distance  at  which  it  acts  from  the  fulcrum. 

For  if  the  magnitudes  of  the  forces  acting  at  A  and  B  are  rep- 
resented by  P  and  W  respectively,  then  (since  P :  W ::  BC  :  AC) 
PxAC=WxBC;  /.PxAC  represents  the  effect  of  P,  and  W 
xBC  represents  the  effect  of  W,  to  turn  the  lever  round  C. 

*  For  since  the  equilibrium  does  not  at  all  depend  upon  the  length  of  the  lines  AP 
BW,  we  may  suppose  those  lines  to  vanish ;  in  which  case  the  centers  of  gravity  of 
P,  W,  may  be  considered  as  coinciding  with  the  extremities  A,  B,  of  the  lever. 


NATURAL   PHILOSOPHY. 


101.  Any  number  of  weights  will  keep  each  other  in  er/uilibrio 
upon  the  arms  of  a  straight  lever,  when  the  sums  of  the  products 
arising  from  multiplying  each  weight  by  its  distance  from  the  ful- 
crum are  equal  on  the  two  sides  of  that  center. 


Fig.  65. 
ABC  D 


Let  AD  (Fig.  65,)  represent  a  straight  lever  whose  fulcrum  is 
G,  and  let  the  bodies  or  weights  A,  B,  C,  D,  be  placed  upon  its 
arms  AG,  DG,  at  different  distances  from  G  ;  then  the  effort 
of  A  to  turn  the  lever  about  G  being  represented  by  AxAG,  of 
B  by  BxBG,  of  C  by  C  xCG,  &c.,  the  whole  effect  upon  the  arm 
AG  will  be  represented  by  AxAG+BxBG,  and  upon  the  arm 
DG  by  C  xCG+D  xDG  ;  there  will  consequently  be  an  equilibrium 
when  AxAG+BxBG=CxCG+DxDG. 

102.  Levers  are  divided  into  three  different  orders,  according 
to  the  position  of  the  power  and  weight  with  respect  to  the  ful- 
crum. I.  In  a  lever  of  the  first  order,  the  fulcrum  is  between 
the  power  and  weight,  as  in  the  preceding  instance  ;  and  here 
the  pressure  on  the  fulcrum  is  equal  to  the  sum  of  the  weights. 
II.  In  a  lever  of  the  second  order,  the  weight  is  placed  between 
the  power  and  the  fulcrum,  as  in  the  annexed  figure,  where  the 
weight  W  is  supported  by  the  power  P  acting  upward  in  the  di- 
rection AP.  In  this  case  also  there  is  an  equilibrium,  when  the 
power  and  weight  are  inversely  as  the  arms  on  which  they  re- 
spectively act  ;  for  the  effort  of  the  weight  W  to  turn  the  lever 
about  C  is  measured  by  WxBC  (Art.  68,  Note  ;)  the  effort  of  the 
power  P  (acting  in  the  direction  PA)  to  turn  the  lever  about  C, 
Fig.  66. 


i.  e.  to  sustain  W,  is  measured  by  PxAC ;  when  there  is  an  equi- 
librium, therefore,  PxAC  must  be  equal  to  WxBC,  or  P:W 
: :  BC :  AC,  as  before.  Therefore,  P  is  less  than  W ;  and  the  pres- 
sure upon  the  fulcrum  (P  and  W  acting  in  opposite  directions)  is 


MECHANICS.  89 

equal  to  W — P ;  for  the  pressure  at  A  is  the  same  as  would 
be  exerted  on  a  fulcrum  at  that  point,  in  which  case  the  pressure 
on  both  points  C  and  A  would  equal  the  whole  weight  W ;  there- 
fore, the  pressure  on  C  equals  W— P.  III.  In  a  lever  of  the  third 
order,  the  power  acts  between  the  weight  and  the  fulcrum ;  but 
the  equilibrium  is  produced  on  the  same  principle  as  before ;  for 
an  equilibrium  will  take  place  when  the  opposite  forces  P  and 
W  are  equal ;  which  will  be  when  PxAC=WxBC  ;  or  when 
Fig.  67. 


8 


P :  W : :  BC :  AC.  Since  BC  is  greater  than  AC,  P  is  greater  than 
W,  and  the  pressure  upward  from  the  fulcrum  is  represented  by 
P— W.*  Hence  we  have  the  following  general  principle  appli- 
cable to  the  three  orders  of  levers, 

When  the  forces  act  PERPENDICULARLY  TO  THE  ARMS  OF  A  STRAIGHT 
LEVER,  an  equilibrium  is  produced,  if  the  power  is  to  the  weight  as 
the  distance  of  the  weight  from  the  prop  is  to  the  distance  of  the 
power  from  the  prop. 

In  the  second  kind  of  lever  the  weight  is  greater  than  the 
power  ;  in  the  third  kind,  less. 

When  a  weight  is  sustained  between  two  props,  the  part  sustained 
by  each  prop  is  inversely  as  the  distance  of  the  weight  from  it. 

Fig.  68. 
C  Be 


I 


For  suppose  C,  c  to  be  successively  the  centers  of  motion,  then 
Press,  on  fulcrum/ :  weight  W : :  BC :  Cc ;  for  same  reason, 
Weight  W  :  press,  on  fulcrum  F : :  Cc  :  Be  ;  .'. 
Press,  on  fulcrum/:  press,  on  fulcrum  F ::  BC  :  Be  ;  and  as  the 
whole  weight  is  sustained  by  the  two  props,  it  is  divided  between 
them  in  the  ratio  of  BC  :  Be. 

*  In  this  third  order  of  levers,  although  the  lever  is  supposed  to  move  freely  round 
the  center  of  motion  C,  it  is  yet  necessary  to  consider  it  as  firmly  connected  with  the 
prop  at  that  point. 

12 


90  NATURAL   PHILOSOPHY. 

103.  Let  us  next  estimate  the  relation  of  the  forces  which  keep 
a  lever  in  equilibrio,  when  its  own  weight  is  taken  into  consider- 
ation. Since  this  weight  may  be  considered  as  collected  in  the 
center  of  gravity  of  the  lever,  (Art.  66,)  its  effective  force  is  equal 
to  the  weight,  multiplied  into  the  distance  of  its  center  of  gravity 
from  the  fulcrum.  (Art.  100.)  Suppose  the  lever  to  be  of  a 
cylindrical  or  prismatic  form,  and  that  its  weight=iw,  then, 

In  a  lever  of  the  first  order,  (Fig.  64,)  since  the  center  of  grav- 
ity of  the  lever  is  in  the  middle  point,  and  the  distance  of  this 
point  from  the  fulcrum  =i  (AC—  BC),  the  effect  of  the  weight 
of  the  lever  =%w  (AC—  BC),  and  is  exerted  in  the  direction  of 
the  longer  arm  AC.  In  the  case  of  an  equilibrium,  therefore, 
we  have, 

PxAC+£M>(AC—  BC)=WxBC, 

or  p-WxBC-^(AC~BC) 

AC 

In  a  lever  of  the  second  order,  (Fig.  66,)  the  whole  weight  of 
the  lever  operates  in  conjunction  with  W,  and  the  distance  of 
the  center  of  gravity  from  the  fulcrum  in  this  case  =£AC,  .*. 
PxAC=WxBC+^xAC, 
,     WxBC 


In  a  lever  of  the  third  order,  (Fig.  67,)  the  whole  weight  of  the 
lever  operates  in  conjunction  with  W,  and  the  distance  of  the 
center  of  gravity  from  the  fulcrum  =5BC,  .•. 
PxAC=WxBCH-Jtt»xBG, 
np  p_(W+lu>)xBC 

"~~AC~~ 

We  have  thus  far  confined  our  attention  to  the  case  in  which 
the  lever  is  supposed  to  be  straight,  and  the  forces  to  be  applied 
at  right  angles  to  it  ;  we  now  propose  to  take  a  more  general 
view  of  the  properties  of  the  lever,  whatever  be  its  shape,  or  the 
directions  of  its  forces. 

104.  Two  forces  acting  at  the  extremities  of  the  arms  of  ANY 
lever  will  be  in  equilibrio,  when  they  are  to  each  other  inversely  as 
the  perpendiculars  let  fall  upon  the  lines  of  direction  in  which  they 
respectively  act.* 

Let  ACB  (Fig.  69)  be  any  lever  whose  fulcrum  is  C  ;  and  let 
two  forces,  P,  p,  act  in  the  directions  AP,  Bp,  upon  the  extremi- 
ties of  its  arms  CA,  CB.  Produce  PA,  pQ,  to  M,  N,  and  let  fall 
the  perpendiculars  CM,  CN  ;  with  the  longer  perpendicular  CN 
and  center  C  describe  the  circular  arc  ND,  and  join  CD.  Let 
DP  represent  the  magnitude  of  the  force  acting  upon  the  lever 
at  A  in  direction  AP,  and  let  it  be  resolved  into  two  others,  viz. 

*  This  theorem  evidently  embraces  the  proposition  in  Art.  99. 


91 


DE  perpendicular  and  EP  parallel  to  the  radius  CD  ;  then  DE 
only  is  effectual  to  produce  motion  round  the  center  C,  the  part 
EP  being  exerted  merely  to  produce  pressure  upon  the  fulcrum 
in  direction  CD.  Supposing,  therefore,  the  lever  to  be  perfectly 
inflexible,  this  effort  of  P  to  produce  motion  round  C  would  be 
counteracted  by  a  force  equal  to  DE,  applied  perpendicularly  at 
N,  in  direction  NB.  Let  then  the  power  p  be  equal  to  that  part 
of  P  which  is  represented  by  DE,  and  (as  it  is  indifferent  in  what 
point  of  the  line  of  direction  this  power  acts)  conceive  it  to  act 
at  N.  In  this  case  the  forces  P,  p  will  be  in  equilibrio ;  i.  e.  when 
P  :p  : :  PD  :  DE,  the  lever  will  be  kept  in  equilibrio  about  the 
center  of  motion  C  ;  but  by  similar  triangles 

PD  :  DE  : :  CD  (CN) :  CM, .-.  P  : p  : :  CN  :  CM. 

105.  Produce  PA,  pE  (Fig.  70)  till  they  meet  in  S ;  join  CS, 
and  draw  CO  parallel  to  pS,  in  which  case  the  angle  OCS=CSN. 
Now  if  CS  is  made  radius,  CM  becomes  the  sine  of  CSO,  and 
CN  the  sine  of  CSN  or  OCS ;  but  as  SO :  OC : :  sin.  OCS(CN) : 
sin.  CSO(CM ;)  hence  P  :  p : :  (CN  :  CM  : :)  SO  :  OC.  The  two 
sides  SO,  OC  of  the  triangle  SOC  represent,  therefore,  the  rela- 
tive magnitude  and  direction  of  the  two  forces  P,  p ;  the  third 
S  Fig.  70. 


side  SC  will  consequently  represent  a  force  equivalent  to  them 
both,  (Art.  43 ;)  and  as  this  compound  force  acts  directly  toward 


NATURAL   PHILOSOPHY. 


C,  the  pressure  upon  the  fulcrum  will  be  represented,  in  quantity 
and  direction,  by  the  line  SC. 


N 


106.  If  each  arm  of  the  lever  is  M       Fig.  71. 

straight,  (Fig.  71,)  but  the  two 
arms  are  inclined  to  each  other  in 
the  given  angle  ACB,  let  CAM=a, 
CBN=ft  and  rad.=l ;  then 

AC:CM::rad.  (l):sin.  a, 
.*.  CM=ACx  sin.  a,  and 

BC:CN::rad.  (l):sin.  ft 

.•CN=BCxsin.  ft     Hence 
P  :p : :  BC  x  sin.  /8  :  AC  x  sin.  a. 
If  sin.  a=sin.  ft  then  P  :p  : :  BC : 

AC,  or  P, ,p,  are  to  each  other  inversely  as  the  arms  of  the  lever 
upon  which  they  respectively  act ;  which  shows  that  the  same 
law  of  equilibrium  obtains  in  the  bent  as  in  the  straight  lever, 
when  the  forces  act  at  equal  angles. 


AM 


Fig.  72. 
C 


107.  If  the  lever  is  straight,  (Fig. 
72,)  and  the  forces  act  parallel  to 
each  other,  then  sin.  a=sin.  ft, 

.'.  P:W::BC:AC, 
as  in  Art.  106 ;  and  this  will  be  the 
case  whatever  be  the  position  of  the 
lever;  if  therefore  P  and  W  are  in 
equilibrio  when  the  lever  is  in  the 
horizontal  position  ACB,  they  will  also 
be  in  equilibrio  when  it  is  in  any  other 

position  aCb ;  i.  e.  the  lever  thus  acted  upon  will  rest  in  any 
position.* 


W 


w 


108.  In  the  common  balance  or 
scales,  the  arms  AC,  CB,  (Fig.  73,) 
are  equal  to  each  other ;  /.  when 
there  is  an  equilibrium,  P=W.  But 
this  equilibrium  will  be  destroyed, 
if  either  P  or  W  is  removed  from 
its  perpendicular  position.  Sup- 
pose, for  instance,  a  person  placed 
in  the  scale  P  is  balanced  by  the 
weight  W,  but  by  pushing  in  the 


Fig.  73. 
F    C 


W 


*  This  also  appears  from  Art.  104 ;  for  produce  Pa  to  M,  then,  when  the  lever  is  in 
position  aCb,  P :  W : :  CN  :  CM  : :  (by  sim.  triangles)  Cb  :  Ca : :  (since  the  arms  of  the 
levet  are  invariable)  CB  :  CA.  In  thus  asserting  that  the  lever  will  rest  in  any  posi- 
tion, it  is>  of  course  taken  for  granted,  that  the  common  center  of  gravity  of  P,  W, 
and  the  lever,  coincides  with  the  center  of  motion  ;  for  it  is  evident,  from  the  princi- 
ples laid  down  in  Chap.  IV,  that  the  lever  will  only  rest  when  that  centejr  is  supported 


MECHANICS.  93 

oblique  direction  pF  against  the  arm  CA,  the  scale  is  protruded 
into  the  position/) A  ;  then  draw pE  parallel  to  PA,  and  produce 
CA  to  meet  it  in  E ;  and  at  the  instant  the  scales  arrive  at  the 
position  p A,  the  power  will  act  at  the  perpendicular  distance 
CE  from  the  center  of  motion  ;  its  effect,  therefore,  (by  Art.  100,) 
to  turn  the  lever  about  C,  will  be  measured  by  PxCE,  i.  e.  its 
effect  will  exceed  that  of  W  in  the  proportion  of  CE  :  CA,  and 
consequently  the  scale  in  which  the  person  is  will  preponderate, 
and  the  equilibrium  be  destroyed. 

109.  In  the  COMPOUND  LEVER,  the  opposite  forces  are  in  equilibria, 
when  the  power  is  to  the  weight,  as  the  product  of  all  the  arms  on  the 
side  of  the  weight,  is  to  the  product  of  all  the  arms  on  the  side  of  the 
power. 

In  a  combination  of  levers  connected  with  each  other  in  the 
manner  represented  in  the  annexed  figure,  (Fig.  74,)  there  will  be 
Q  Fig.  74. 

V         D  G     E 


an  equilibrium  when  P:W::BCxDFxEG:ACxBFx DG.  For 
suppose  the  equilibrium  to  exist,  and  that  the  forces  which  act  at 
B,  D,  are  represented  by  Q,  R,  respectively,  then 


Q:R::DF:BF, 


DG 
/.P:W::BCxDFxEG:ACxBFxDG.  (A.) 

Also,  the  pressure  on  C=P-f  Q= 


. 
DG 

We  have  here  supposed  the  forces  to  act  perpendicularly  to 
the  extremities  of  the  several  levers  ;  if  they  acted  obliquely,  or 
if  the  arms  of  the  levers  were  inclined  to  each  other,  then  for 
these  arms  must  be  substituted,  in  the  proportion  marked  (A,)  the 
perpendiculars  let  fall  from  the  centers  of  motion,  C,  F,  G,  upon 
the  lines  of  direction  in  which  the  forces  act.  We  now  proceed 
to  illustrate  the  foregoing  theory  by  a  few  plain  examples. 


4  NATURAL   PHILOSOPHY. 

110.  EXAMPLES. 

1.  At  the  extremities  of  a  straight  lever,  whose  length  is  24 
inches,  are  placed  two  weights  of  5  and  7  pounds :  At  what 
point  must  the  fulcrum  be  placed  so  that  these  weights  shall 
balance  each  other,  the  weight  of  the  lever  not  being  taken  into 
the  account? 

This  is  the  case  of  the  lever  of  the  first  order,  in  which  (Fig. 
64)  P=5,  W=7,  and  AB=24  ;  and  when  there  is  an  equilibrium, 
P(5):W(7)::BC:AC;.\12:7::BC+AC(AB=24):AC=VV=14; 
hence  BC=24— 14=10  inches. 

2.  At  the  extremity  of  a  lever  of  the  second  order,  there  acts 
a  power  which  is  of  itself  able  to  sustain  only  a  weight  of  15 
pounds  ;  but  when  acting  under  this  mechanical  advantage,  it  is 
able  to  sustain  a  weight  of  100  pounds,  placed  five  feet  from  it : 
What  is  the  length  of  the  lever  ? 

Referring  to  Fig.  66, 100  :  15  : :  AC :  AC— 5, .-.  100 : 85  : :  AC :  5, 

.'.  AC=^=5if  feet, 
oo 

3.  A  body  suspended  at  the  extremities  of  a  balance  whose 
arms  are  unequal,  weighs  p  pounds  at  one  end,  and  q  pounds  at 
the  other :  What  is  its  real  weight  ? 

A  balance  of  this  kind  is  called  a  false  balance,  because  when 
the  body  is  suspended  at  the  extremity  of  the  longer  arm,  the 
weight  which  balances  it,  is  above,  and  when  suspended  at  the 
extremity  of  the  shorter  arm,  is  below  the  true  weight.  But  the 
true  weight  of  the  body  is  easily  found  by  the  following  opera- 
tion ;  viz.  Let  x=ihe  true  weight,  and  a  the  arm  of  the  lever 
upon  which  p  is  suspended  to  balance  it ;  and  b  the  arm  upon 

which  q  is  suspended  to  balance  it ;  then  x  :p  : :  a  :  b,  or  x=  2? 
and  x  :  q : :  b :  a  or  x—^-\  multiply  these  two  equations  together 

and  we  have  x3=pq  or  x=Vpqi  i.  e.  the  true  weight  of  the  body 
is  a  mean  proportional  between  the  apparent  weights  thus  ob- 
tained. Hence  to  find  the  weight  of  a  body  by  a  false  balance, 
we  have  this 

RULE. — Take  the  weight  of  the  body  in  each  scale  ;  multiply 
together  the  two  weights  thus  found,  and  take  the  square  root  of  the 
^product. 

4.  To  explain  the  construction  of  the  steelyard. 

A  steelyard  is  a  lever  of  the  first  order,  having  two  unequal 


MECHANICS.  95 

arms  BC,  CD,  (Fig.  75 ;)  a  given  weight  P  is  movdble  along  the 
longer  arm  CD,  so  as  to  sustain  a  weight  of  a  variable  mag- 
nitude suspended  from  the  extremity  of  a  shorter  arm  CB. 
Fig.  75. 


C  E 


•1 


1.  Suppose  the  weight  of  the  arms  not  to  be  taken  into  con- 
sideration, then  P :  W : :  BC  :  AC  (Art.  99,)  .-.  W  xBC=P  x  AC,  and 
as  P  and  BC  are  given,  Woe  AC.     Hence  if  C«,  ab,  be,  &c.,  be 
taken  equal  to  each  other,  (or  C6=2Ca,  Cc— 3Ca,  &c.,)  then  if  P 
balances  one  pound  when  placed  at  a,  it  will  balance  two  pounds 
at  b,  three  pounds  at  c,  &c. 

2.  Let  us  next  suppose  the  steelyard  to  have  weight,  and  that 
the  excess  of  the  weight  of  the  longer  arm  CD  above  that  of  the 
shorter  CB,  is  such  that  the  movable  weight  P,  when  placed  at 
E,  would  keep  the  arms  in  equilibrio ;  in  which  case  this  excess 
would  be  measured  by  PxCE;  if  therefore  a  weight  W,  placed 
at  B,  be  in  equilibrio  with  the  weight  P,  placed  at  A,  we  should 
have  WxBC=PxAC+PxCE=P  (AC+CE)   =PxAE;    conse- 
quently, since  P  and  BC  are  given,  Wx  AE.     The  construction 
of  the  steelyard,  therefore,  would  be  the  same  as  in  the  former 
instance,  except  that  the  graduation  must  begin  from  E,  instead 
of  from  C. 

If  the  longer  arm  be  divided  into  equal  parts,  to  indicate  the 
number  of  pounds,  ounces,  &c.,  which  are  contained  in  the  vari- 
able weight  W,  the  magnitude  of  the  divisions  may  be  found. 

AF     RP 
For  WxBC^PxAE  ;  .'.  ^=^,  and  as  AE  is  the  extent  of  the 

A  "P         "RP 

graduated  arm  corresponding  to  W,  -v^or  -^  will  be  the  length 

of  a  division  corresponding  to  1  pound  or  1  ounce,  &c. 

Hence,  when  only  BC  and  P  are  given,  the  magnitude  of  each 

T>/"i 

division  will  be  equal  to  — . 

5.  ACB  is  a  cylindrical  straight  lever  whose  weight  is  (w)  at 
whose  extremity  A,  a  given  weight  (P)  is  suspended:  It  is  required 
to  determine  the  position  of  the  fulcrum  C,  so  that  P  may  be  in 
equilibrio  with  the  longer  arm  BC. 

LetAB=«,                                     B           Fig.  76.  C         A 

AC=x,  " — T" 

Then  the  distance  of  the  center 

of  gravity  of  the  lever  from  the  p 

fulcrum— \a— x. 


96 


NATURAL   PHILOSOPHY. 


Hence,  from  the  principles  laid  down  in  Art.  103,  when  there 
s  an  equilibrium,  we  have 

'=w($a— x),  or  2Px-}-2wx=wa,  and  x= 


Cor.  If  P=MJ,  then  x=ia,  or  AC=iAB,  the  length  of  the  lever. 

6.  P  and  W  are  suspended  from  the  extremities  of  the  arms 
of  the  bent  lever  ABC,  (whose  weight  is  not  taken  into  consid- 
eration :)  It  is  required  to  find  the  angle  of  inclination  (ACB,) 
so  that  when  there  is  an  equilibrium,  AC  shall  be  parallel  to  the 
horizon. 


By  Art.  106, 

P  :  W  :  :  BC  xsin.  CBD  :  AC  xsin.  PAC  ;  A 
but  sin.  PAC=rad.=l,andsin.CBD=cos. 
BCD  ;  .'.  P  :  W  :  :  BC  xcos.  BCD  :  AC  x  1, 

PxAC 
hence  cos.  BCD=  >   from  which 


Fig.  77. 


BCD  and  consequently  ACB  is  known. 

AC 


Cor.  1.  If  P=W,then  cos.  BCD; 


BC 


and  if  BC=2AC,  then  gg=£  ; 

.-.  cos.  BCD=£=cos.  60°  ;  hence  ACB=120°. 

Cor.  2.  If  AC=BC,  then  cos.  BCD=^  ;  and  if  W=2P,  then 

p 

•yy=£  ;  /.  cos.  BCD=£=cos.  60°  ;  hence,  in  this  case  also,  ACB 

=120°. 

7.  From  the  extremities  of  the  arms  CA,  CB,  of  a  bent  lever, 
the  weights  P,  W,  are  suspended :  It  is  required  to  determine 
the  position  of  the  lever  when  these  weights  are  in  equilibrio. 

Through  C  (Fig.  78,)  draw  MN  parallel  to  the  horizon,  and 
produce  PA,  WB  to  meet  it  in  M  and  N  ;  by  Art.  104,  the  lever 
will  be  in  equilibrio,  when  Fig.  78. 

P  :  W  : :  CN  :  CM. 

Join  AB,  and  draw  CD  parallel  to  MP 
or  NW ;  CD  will  cut  AB  in  the  same  A 
ratio  that  it  does  MN,*  i.  e. 

DB  :  DA  : :  CN  :  CM  ; 
hence  DB  :  DA  : :  P  :  W  ;   from  which 
it  appears  that  if  the  line  AB,  which 
joins  the  extremities  of  the  arms  of  the 
lever,  be  divided  in  the  point  D,  in  the  ratio  of  P  :  W,  and  that 


MECHANICS.  97 

point  be  brought  immediately  under  the  center  of  motion,  it  will 
give  the  position  of  the  lever  when  P  and  W  are  in  equilibrio. 

111.  QUESTIONS  UPON  THE  PRINCIPLES  OF  THE  LEVER. 

1.  At  one  extremity  of  a  straight  lever  whose  length  is  7  feet, 
a  weight  of  10  pounds  is  suspended ;  at  the  distance  of  5  feet 
from  the  point  of  suspension  a  fulcrum  is  placed  :    What  weight 
must  be  suspended  from  the  other  extremity  of  the  lever,  to  keep 
it  in  equilibrio  ?  Ans.  25  pounds. 

2.  A  lever  of  the  second  order  is  25  feet  long  :     At  what  dis- 
tance from  the  fulcrum  must  a  weight  of  125  pounds  be  placed, 
so  that  it  may  be  supported  by  a  power  able  to  sustain  60  pounds, 
acting  at  the  extremity  of  the  lever  1  Ans.  12  feet. 

3.  A  cylindrical  straight  lever  is  14  feet  long,  and  weighs  6 
Ibs.  5  oz.  ;  its  longer  arm  is  9,  and  its  shorter  5  feet ;  at  the  ex- 
tremity of  its  shorter  arm  a  weight  of  15  Ibs.  2  oz.  is  suspended: 
What  weight  must  be  placed  at  the  extremity  of  the  longer  arm 
to  keep  it  in  equilibrio  ?     (See  Art.  103.)  Ans.  7  pounds. 

4.  A  body  weighs  1 1  pounds  at  one  end  of  a  false  balance, 
and  17  Ibs.  3  oz.  at  the  other :  What  is  its  real  weight  ? 

Ans.  13  Ibs.  and  12  oz. 

5.  A  and  B  are  of  the  same  height,  and  sustain  upon  their 
shoulders  a  weight  of  150  pounds,  placed  on  a  pole  9£  feet  long; 
the  weight  is  placed  6|  feet  from  A :  What  is  the  weight  sus- 
tained by  each  person  ?     (See  Art.  102.) 

Ans.  A  sustains  42%  pounds,  and  B  sustains  107}  pounds. 

6.  The  longer  arm  of  a  steelyard  is  2  feet  2  inches  in  length, 
and  the  shorter  2|  inches  ;  and  its  apparatus  of  hooks,  &c.,  is  so 
contrived,  that  a  weight  of  two  pounds  placed  upon  the  longer 
arm,  at  the  distance  of  10  inches  from  the  center  of  motion,  will 
balance  8  pounds  placed  at  the  extremity  of  the  shorter  arm ; 
the  movable  weight  (of  2  pounds)  cannot  conveniently  be  placed 
nearer  to  the  fulcrum  than  f  of  an  inch  :    What  must  be  the 
graduation  of  the  steelyard  that  it  may  weigh  ounces,  and  what 
will  be  the  greatest  and  least  weights  that  can  be  ascertained 
by  it? 

Ans.   The  graduation  is  to  12ths  of  an  inch  ;  and  it  will  weigh 
from  1  to  20  pounds. 

7.  The  arms  of  a  straight  lever  are  to  each  other  as  7  :  9,  and 
it  is  acted  upon  obliquely  by  two  forces  ;  the  force  (P)  applied 
at  the  extremity  of  the  longer  arm,  is  inclined  to  it  at  an  angle 
of  50°,  and  (p)  at  the  shorter  at  an  angle  of  80°  :   What  is  the 
proportion  between  the  forces,  when  the  lever  is  in  equilibrio  ? 
(See  Art.  106.)  Ans.  P=p. 

8.  The  arms  of  a  bent  lever  are  equal,  and  P  :  W : :  1  :  2 : 
What  must  be  the  inclination  of  the  arms  to  each  other,  that  the 
arm  from  whose  extremity  P  is  suspended  may  be  parallel  to  the 
horizon?  Ans.  120°. 

13 


98 


NATURAL   PHILOSOPHY. 


9.  In  a  combination  of  levers  connected  together  in  the  man- 
ner represented  in  Fig.  74,  the  three  shorter  arms  (BC,  DF,  EG,) 
are  respectively  2,  5,  and  3  feet ;  the  three  longer  arms  (AC,  BF, 
DG,)  are  13,  14,  and  15  feet ;  the  weight  (P)  suspended  from  A 
is  5  pounds :  What  weight  will  it  sustain  at  E  ? 

Ans. 


CHAPTER  VII. 
OF  THE  WHEEL  AND  AXLE ;  AND  THE  PULLEY. 

WHEEL   AND    AXLE. 

112.  IN  order  to  explain  the  manner  in  which  the  wheel  and 
axle  operate  upon  each  other,  suppose  DE  (Fig.  79,)  to  be  a  cy- 
lindric  roller  supported  upon  the  props  LH,  MQ,  and  movable 
about  the  axis  LM.  Let  two  straight  inflexible  rods  AG,  BC  be 
inserted  into  this  cylinder,  in  a  direction  perpendicular  to  the  axis, 
Fig.  79.  P 


L   ii 


but  parallel  to  each  other  and  the  horizon ;  let  there  be  another 
rod  PK  perpendicular  to  the  axis,  but  making  any  angle  with  the 
plane  passing  through  BC  or  AG,  and  the  axis.  From  the  ex- 
tremities of  the  rods  BC,  AG,  let  the  weights  W,  w  be  suspended ; 
then  (Art.  100)  WxBC  will  represent  the  effect  of  W,  and  wx 
AG  the  effect  of  w,  to  turn  the  roller  about  the  axis  LM  ;  and 
supposing  the  rods  AG,  BC,  and  the  roller  DE,  to  be  perfectly 
rigid  and  inflexible,  it  is  evident  that  these  two  weights  will  coun- 
teract each  other's  effects,  in  the  same  manner  as  if  they  were 
acting  at  the  extremities  of  the  arms  of  a  straight  lever.  When 


MECHANICS. 


99 


W  and  w  therefore  arc  in  equilibrio,  WxBC  will  be  equal  to  w 
xAG,  or  w  :  W  : :  BC  :  AG ;  and  this  will  be  the  case  whatever 
be  the  length  of  the  rod  BC.  Suppose  that  rod  to  be  equal  to 
the  radius  of  the  roller,  then  the  string  BW  will  become  a  tangent 
to  the  roller,  and  the  foregoing  proportion  becomes  w  :  W  : :  the 
radius  of  the  roller  :  AG.  Let  us  next  suppose  the  weight  W  to 
be  kept  in  equilibrio  by  a  power  P  acting  at  right  angles  to  the 
extremity  of  the  rod  PK  ;  then  may  P  and  W  be  considered  as 
acting  at  the  extremities  of  the  arms  PK,  BC  of  the  bent  lever ; 
and  since  they  act  at  right  angles  to  those  arms,  (Art.  106)  P  : 
W  : :  BC  :  PK  ;*  and  (when  B  W  acts  as  a  tangent  to  the  roller) 
P  •  W  : :  radius  of  the  roller  :  PK. 

113.  In  the  wheel  and  axle  an  equilibrium  is  produced  when  the 
power  acting  at  the  circumference  of  the  wheel  :  weight  sustained 
upon  the  axle  : :  radius  of  the  axle  :  the  radius  of  the  wheel,  f 
Fig.  80. 

N  T' 


H  Q 

For  let  the  weight  W  (Fig.  80)  (which  is  suspended  by  a  rope 
going  round  the  axle  DE)  be  kept  in  equilibrio,  either  by  another 
weight  (w)  suspended  from  a  rope  going  freely  round  the  wheel 
NAO,  or  by  a  power  P  acting  at  right  angles  to  the  handles  TS, 
Pt,  &c.  of  the  wheel  StV,  and  let  the  planes  of  these  wheels  be 
at  right  angles  to  the  axis  LM  of  the  machine  ;  then,  in  the  for- 

*  For  in  this  case  sin.  a=sin.  /3. 

t  Let  R=radius  of  the  wheel,  r=radius  of  the  axle,  then  P  :  W  : :  r  :  R,  .'.Px 

K=Wxr ;  if  W  and  r  be  given,  and  P  and  R  variable,  then  Poc—  ;  i.  e.  to  sustain 

a  given  weight  upon  a  given  axle,  the  power  must  be  increased  as  the  radius  of  the 
wheel  is  diminished  j  and  vice  versa. 


100 


NATURAL   PHILOSOPHY. 


mer  case,  the  weight  W  may  be  considered  as  sustained  by  a 
weight  (w)  acting  at  right  angles  to  the  extremity  of  the  arm  AC 
of  a  straight  lever,  and  in  the  latter,  by  a  power  (P)  acting  at 
right  angles  to  the  extremity  of  the  arm  ^K  of  a  bent  lever,  the 
weight  itself  in  each  case  acting  at  the  distance  of  the  radius  of 
the  axle  from  the  center  of  motion  ;  hence,  (by  Art.  112,) 
w  or  P  :  W  : :  radius  of  the  axle  :  AC  or  iK. 

1 14.  If  the  power  does  not  act  _  Fig.  81. 
at  right  angles  to  the  radius  of 

the  wheel,  but  in  some  oblique 
direction,  as  AP  (in  the  annexed 
figure,  which  represents  a  sec- 
tion of  the  wheel  and  axle,)  then 
let  fall  CD  perpendicular  to  AP. 
By  the  property  of  the  lever,  P 
and  W  are  to  each  other  inversely 
as  the  perpendiculars  let  fall  from 
the  center  of  motion  upon  the 
lines  of  direction  in  which  they 
respectively  act,  (Art.  104 ;).  in 
this  case,  therefore, 
P  :  W  : :  CB  :  CD  : :  radius  of  the 
axle  :  radius  of  the  wheel  x  sine  of 
the  angle  which  P  makes  with  the 
radius  of  the  wheel. 

115.  Hitherto  we  have  not  considered  the  thickness  of  the 
rope  ;  when  that  is  taken  into  the  account,  we  must  add  the  half 
of  it  to  the  distance  at  which  W  and  w  respectively  act.*     Let 
therefore  2Z=diameter  of  the  rope,  and  let  R— rad.  of  the  wheel, 
r=rad.  of  the  axle  ;  then  if  the  thickness  of  the  rope  be  taken 
into  consideration,  we  have  w  :  W  : :  r-\-t  :  R-H ;  and  since  in 
this  case  the  same  quantity  (t)  is  added  to  each  term  of  the  ratio 
7-  :  R,  w  must  bear  a  greater  ratio  to  W  than  that  of  r  :  R,  or  of 
the  radius  of  the  axle  to  the  radius  of  the  wheel.f 

116.  In  a  combination  of  wheels,  such  as  is  represented  in 
figure  82,  where  a  power  (P)  acts  upon  the  winch  or  handle 
PQ,t  which  turns  the  wheel  A,  which  acts  upon  the  wheel 

*  For  W,  w,  act  in  the  direction  of  the  axis  of  the  rope,  and  this  axis  is  evidently 
removed  from  the  circumference  of  the  wheel  or  axle  by  half  the  thickness  of  the  rope. 

t  In  this  article  we  have  considered  the  ropes  which  go  round  the  wheel  and  axle  to 
be  of  the  same  thickness,  and  that  the  rope  coils  round  the  axle  but  once.  But  suppose 
the  thickness  of  the  rope  to  which  W  is  appended  to  be  2T,  that  of  w  to  be  2t,  and 
the  rope  to  coil  round  the  axle  any  number  of  times  denoted  by  n  ;  then  it  is  evident, 
that  for  each  coil  of  the  rope  after  the  first,  W  will  be  further  removed  from  the  cir- 
cumference of  the  axle  by  the  whole  thickness  (2T)  of  the  rope;  the  most  general 
form  therefore  under  which  the  relation  of  w  :  W  can  be  exhibited,  when  the  thick- 
ness of  the  rope  is  taken  into  consideration,  is  w  :  W  : :  r-f-(2w— 1)  T  :  R-f*. 

t  In  this  case  the  effect  will  evidently  be  the  same  as  if  the  power  acted  at  th« 
circumference  of  a  wheel  whose  radius  is  PQ. 


101 


B,  from  which  the  motion  is  propagated  through  the  wheels  C 
and  D  to  the  axle  E,  about  which  the  rope  that  sustains  the 
weight  (W)  is  wound,  let  the  force  exerted  by  the  wheel  A  upon 
the  wheel  B=/>,  by  C  upon  D=q ;  then,  supposing  P  and  W  to 
be  in  equilibrio,  we  have, 

P  :  p  : :  rad.  of  wheel  A  (r)  :  PQ, 

p  :  q  : :  rad.  of  wheel  C  (r')  :  rad.  of  wheel  B  (R), 

g  ;  W  ; ;  rad.  of  axle       (a)  :  rad.  of  wheel  D  (R') ; 

.-.P:  W::axrxr'  :  PQxRxR';* 

where  the  demonstration  goes  upon  the  same  principle  as  that 
of  a  combination  of  levers,  in  Art.  109. 

117.  Instead  of  the  power  being  applied  to  the  handle  or  winch 
PQ,  (Fig.  82,)  suppose  it  to  be  applied  at  the  circumference  of  a 
wheel  whose  radius  is  (R ;)  let  the  radius  of  the  axle— r,  the  radii 
of  the  small  wheels  =r',  r",  &c.,  and  of  the  larger  ones  =R',  R", 
&c. ;  then,  whatever  be  the  number  of  such  wheels,  the  propor- 
tion expressing  the  relation  between  P  and  W,  when  there  is  an 
equilibrium,  will  be  P :  W  : :  r  xr1  xr"  &c. :  R  xR'  xR"  &c. :: product 
of  the  radii  of  all  the  smaller  wheels  (or  axles) :  product  of  the 
radii  of  all  the  larger  ones. 

If  the  radii  of  the  large  wheels  are  equal  to  each  other,  and 
also  those  of  the  small  wheels,  then  P :  W : :  rn :  Rw,  where  n 
equals  the  number  of  wheels  or  axles. 

118.  EXAMPLES. 

1.  What  must  be  the  diameter  of  a  wheel  by  which  a  weight 
of  100  pounds  suspended  by  a  rope  going  round  an  axle,  whose 


>  Algebra,  Art.  392 


102  NATURAL   PHILOSOPllF. 

radius  is  six  inches,  may  be  kept  in  equilibrio  by  a  power  acting 
upon  it  equivalent  to  12  pounds. 

{then,  by  Art.  113, 
P  (12):W(100)::r  (6):*, 
600  «•  * 

"  X~  "12"  =5°  inches=4     2 
and  the  diameter          8     4 

2.  A  weight  of  500  pounds  is  sustained  by  a  rope  of  one  inch 
diameter,  going  round  an  axle  whose  radius  is  8  inches  ;  and  the 
power  acts  close  to  the  circumference  of  a  wheel  whose  radius 
is  4  feet  :  What  is  the  ratio  of  P  :  W  ? 

This  is  a  case  of  Art.  113,  where  the  weight  is  not  kept  in 
equilibrio  by  another  weight,  but  by  some  power  acting  upon  a 
handle  close  to  the  circumference  of  the  wheel,  (as  P  acts  upon 
the  wheel  StV  in  Fig.  80,)  and  since  t  disappears  in  the  4th  term 
of  the  proportion,  w  (or  P)  :  W  :  :  r+t  :  R+£,  which  becomes 
P  :  W  :  :  r+t  :  R.  In  the  present  instance,  W=500,^=8  inches, 


t=%  inch,  R=48  inches,  .-.  P  :  500  :  :  8*  :  48  ;  or  P=p  =88.54 
pounds.* 

3.  In  Fig.  82,  PQ=1  foot  ;  the  radii  of  the  wheels  A,  C,  are 
each  4  inches  ;  the  radii  of  the  wheels  B,  D  are  each  15  inches  ; 
and  the  radius  of  the  axle  E  is  3  inches  :  What  power  must  be 
applied  to  P  to  support  a  weight  of  600  pounds  ? 

By  Art.  116,  P  :  W  (600)  -.laxrxr1  :  PQxRxR', 
::3x4x4:  12x15x15, 


119.  QUESTIONS  UPON  THK  PRINCIPLES  OP  THE  WHEEL  AND  AXLE. 

1.  A  power  of  14  pounds  acts  upon  a  wheel  whose  diameter 
is  9  feet  :  What  weight  will  keep  it  in  equilibrio,  supposing  the 
rope  which  supports  that  weight  to  be  wound  round  an  axle 
whose  diameter  is  7  inches?  Ans.  216  pounds. 

2.  A  power  of  4  pounds  keeps  in  equilibrio  a  weight  of  176 
pounds,  by  means  of  a  wheel  whose  diameter  is  1  1  feet  :  What 
is  the  diameter  of  the  axle  ?  Ans.  3  inches. 

3.  A  power  (P)  acting  by  means  of  a  rope  going  over  a  wheel 


*  If  the  thickness  of  the  rope  be  not  considered,  then  P  :  W  (500)  :  :  r  (8)  :  R  (48,) 
P=  ^~  =83.33  ;  it  makes  a  diflerenc 
ness  be  or  be  not  taken  into  the  account. 


•.  P=    ~  =83.33  ;  it  makes  a  diflerence,  therefore,  of  5.21  Ibs.,  whether  this  thick- 


MECHANICS. 


103 


whose  diameter  is  7  feet  1 1  inches,  supports  a  weight  of  528 
pounds  ;  the  diameter  of  the  axle  is  9  inches,  and  the  rope  by 
which  P  and  W  are  suspended  is  two  inches  thick :  What  must 
be  the  magnitude  of  P,  supposing  the  thickness  of  the  rope  to  be 
taken  into  consideration  ?  Ans.  P=59.876  pounds. 

4.  Four  wheels,  A,  B,  C,  D,  whose  diameters  are  5,  6,  10,  and 
2  feet  respectively,  are  put  in  motion  by  a  power  of  15  pounds 
applied  at  the  circumference  of  the  wheel  A ;  these  wheels  act 
upon  each  other  by  means  of  three  smaller  wheels,  the  diameter 
of  each  of  which  is  10  inches  ;  the  last  wheel  D  turns  an  axle 
whose  diameter  is  4  inches :  What  weight  may  be  sustained  by 
a  rope  going  over  this  axle  1  Ans.  46,656  pounds. 


THE   PULLEY. 

120.  A  pulley  is  a  small  grooved  wheel  movable  about  a  pivot, 
the  pivot  itself  being  at  the  same  time  either  fixed  or  movable. 
The  principle  upon  which  a  weight  is  sustained  by  means  of  a 
pulley  or  system  of  pulleys,  is  very  simple,  and  will  be  readily 
understood  from  the  following  investigation. 

In  the  single  fixed  pulley  A,  (Fig  83,)  about  which  the  weight 
W  is  sustained  by  the  power  P  acting  on  a  string  WAP  passing 
along  the  groove  in  the  circumference,  no  mechanical  advantage 
is  gained ;  for  since  the  rope  passes  freely  round  the  pulley,  it  is 
evident  that  the  tension  on  each  side  of  it  must  be  the  same,  and 
consequently  that  the  power  must  be  equal  to  the  weight  which 
it  sustains.  The  only  advantage  attending  a  pulley  of  this  kind 
B Fig.  83.  C  B  Fig.  84.  C 

~~' 


W 


is,  that  a  given  power  may  be  made  to  sustain  or  put  in  motion 
a  given  weight  in  a  more  convenient  manner,  by  altering  at 
pleasure  the  direction  in  which  the  power  acts.  The  pressure 
upon  the  pivot  or  axis  of  the  pulley  A  is  evidently  equal  to  P-f  W. 


104 


NATURAL   PHILOSOPHY. 


121.  But  if  a  weight  W  (Fig.  84)  be  sustained  by  a  power  P 
acting  on  a  string  going  over  a  movable  pulley  E  as  well  as  the 
fixed  one  A,  then  it  is  evident  that  this  weight  is  sustained  by 
two  strings  AE,  DE  ;  and  as  it  is  suspended  from  the  center  of 
the  pulley  E,  these  ropes  must  act  at  equal  distances  from  that 
center ;  consequently,  each  string  must  sustain  half  the  weight. 
But  it  is  evident,  that  whatever  is  the  weight  sustained  by  the 
string  AE,  the  same  must  be  sustained  by  the  power  P,  which 
acts  upon  a  string  going  freely  over  the  fixed  pulley  A ;  hence, 
when  there  is  an  equilibrium,  P=£  W,  or  W=2P,  .-.  P :  W : :  1  :  2. 
With  respect  to  the  pressure  upon  the  hook  D,  it  is  $  W  or  P,  and 
upon  the  axis  of  the  pulley  A  it  is  equal  to  P+£W=2P. 

122.  The  same  principle  applies  to  the  system  of  pulleys,  in 
which  the  sam6  string  goes  round  all  the  pulleys,  as  in  Fig.  85. 
For  it  is  evident  that  the  weight  W  is  supported  by  all  the  strings 
at  the  lower  block ;  if  therefore  the  whole  number  of  these  strings 
be  (n,)  each  string  must  support  ^th  part  of  the  weight.     But  when 

B  Fig.  85.  C  Fig.  86. 


there  is  an  equilibrium,  whatever  be  the  tension  upon  each  of 
these  strings  which  support  the  weight,  the  same  will  be  the  ten- 
sion on  the  string  upon  which  the  power  P  acts ;  hence  P=*W, 
or  W=ftP,  .'.  P  :  W  : :  1  :  n*  (where  n=number  of  strings  at  the 

*  If  two  blocks  of  pulleys  of  this  kind  (in  which  m  and  n  are  respectively  the 


105 

lower  block,  or  t.wice  the  number  of  movable  pulleys.)  The 
pressure  upon  the  hook  B  or  C  is  evidently  equal  to  P+W=P+ 
nP=(n+l)P/  ' 

123.  When  the  same  string  does  not  go  round  all  the  pulleys, 
but  each  pulley  (Fig.  86)  has  a  separate  string  CFE,  HDF,  ABD, 
&c.,  going  round  it,  and  fastened  to  the  hooks  A,  H,  C,  &c.,  then 
the  relation  between  P  and  W  must  be  estimated  by  a  different 
method.  Thus  (since  the  string  CFE  goes  over  a  single  movable 
pulley)  by  Art.  121,  P  :  weight  sustained  by  pulley  F  : :  1  :  2; 
and  weight  sustained  by  F  :  weight  sustained  by  D  : :  1  :  2, 

weight  sustained  by  D  :  weight  sustained  by  B,  i.  e.  W  : :  1  :  2, 
.-.  P  :  W  : :  1  :  2x2x2,  &c.,  : :  1  :  2",  or  W=2nP  (if  n  be  the 
number  of  movable  pulleys.)  In  this  system  of  pulleys,  the 
pressure  upon  the  hook  A  =iW=ix2nP=2n-lP  ;  upon  the  hook 
H  (=£  pressure  upon  A)  =£x2n-1P=27l-2P,  &c. ;  and  the  pres- 
sure upon  pulley  E=2P. 

Fig.  87.  C 


124.  Hitherto  we  have  considered  the  strings  as  acting  paral- 
lel to  each  other  ;  but  suppose  the  power  P  (Fig.  87,)  to  act  upon 
the  weight  W  by  a  string  going  over  the  movable  pulley  D  in 
an  oblique  direction  ;  then  produce  the  string  AF  to  E,  and  draw 
DF  at  right  angles  to  DE,  (D  being  the  center  of  the  pulley.) 
Let  FE  represent  the  magnitude  of  the  power  acting  in  the  di- 
rection EF,  which  resolve  into  ED,  DF  ;  then  ED  is  that  part  of 
it  which  is  efficacious  in  supporting  the  weight  W ;  and  since 
the  string  BD  supports  the  same  weight  as  the  string  AF,  the 
whole  weight  sustained  by  the  string  BFA  will  be  represented 
by  2DE  ;  hence, 

P  :  W  : :  EF  :  2DE  : :  rad.  :  2  cos.  DEF  ;  or, 

number  of  strings)  were  combined  together,  so  that  the  effect  (E)  produced  by  the 
first  block  should  act  is  power  upon  the  second,  then  P :  E : :  1 :  m,  and  E  :  W : :  1 :  n : 
.-.  P  :  W  : :  1  :  mn. 

14 


106 


NATURAL    PHILOSOPHY. 


the  power  is  to  the  weight,  as  radius  to  twice  the  cosine  of  the  angle 
of  inclination  of  the  direction  of  the  power  to  that  of  the  weight. 


125.  There  is  another  mode  of  combining 
pulleys  together,  which  we  have  not  yet 
noticed  ;  viz.  when  each  string  is  fixed  into 
the  weight,  as  in  Fig.  88.  In  this  case,  sup- 
posing there  is  an  equilibrium,  and  the  power 
P  acts  upon  a  string  going  freely  over  the 
pulley  A,  then  it  is  evident  that  the  pressure 
upon  that  pulley  will  be  equal  to  2P, /.  the 
string  BA  supports  a  part  of  the  weight 
equal  to  2P.  For  the  same  reason,  since 
the  string  FBA  goes  freely  over  the  pulley 
B,  the  string  CB  supports  4P,  &c. ;  hence 
the  portions  of  weight  supported  by  the 
strings  AG,  BF,  CE,  &c.  are  P,  2P,  4P,  &c. 
respectively,  and  consequently  W=P+2P+ 

4P 2n~1P  (where  n  =  number  of  strings 

attached  to  the   weight)  =  P(l+2+4  .... 
2*-1)=P(2»-l),* 

/.P:  W::l  :  2*-l. 

The  pressure  upon  the  support  at  H  is  evi- 
dently equal  to 

-  1)P=2»P. 


Fig.  88. 
H 


126.  EXAMPLES. 

1.  A  weight  of  56  pounds  is  kept  in  equilibrio  by  a  power  of 
7  pounds,  by  means  of  a  system  of  pulleys,  in  which  the  same 
string  goes  round  every  pulley :  What  is  the  number  of  mova- 
ble pulleys  ? 

Let  n=the  number  of  strings  at  the  lower  block  of  pulleys  ; 

56 

then,  by  Art.  122,  P(7)  :  W(56) : :  1  :  7i=~8=twice  the  num- 
ber of  movable  pulleys,  .*.  the  number  required  is  4. 

2.  In  the  system  of  pulleys  described  in  Art.  123,  find  the  gen- 
eral relation  between  P,  W,  and  n. 

In  this  system,  W=2»P,  .-.  P=; 


*  The  sum  of  a  geometric  series,  whose  first  term  is  1,  common  ratio  2,  and  number 
of  terms  n,  is  2"— 1.    (Alg.  442.) 


MECHANICS.  107 

and  2»=^, 

.'.  n  .  log.  2=\og.  W-log.  P, 

log.  W-log.  P 
or  n=-2—  --  —2  —  . 

log.  2. 

From  which  it  appears,  that  if  any  two  of  the  three  quantities 
P,  W,  and  n  be  given,  the  third  may  be  found. 

3.  Find  the  general  relation  between  P,  W,  and  n,  in  the  sys 
tern  of  pulleys  described  in  Art.  125  ;  and  also  the  number  of 
pulleys  necessary  for  a  power  of  3  pounds,  to  support  a  weight 
of  381  pounds. 


hence  ?ixlog.  2=log.  (W+P)  -log.  P,  or  n=°g'  C+PM°g-  P 


127.  QUESTIONS  ON  THE  PRINCIPLES  OP  THE  PULLEY. 

1.  By  means  of  a  system  of  pulleys,  of  which  five  are  mova- 
ble, and  in  which  the  same  string  goes  round  all  the  pulleys, 
what  power  will  be  necessary  to  sustain  a  weight  of  165  Ibs  ? 

Ans.  IQ^lbs. 

2.  A  weight  is  sustained  by  a  power  attached  to  a  rope  going 
over  one  movable  pulley,  (as  in  Fig.  87,)  the  direction  of  the 
rope  making  an  angle  of  60°  with  a  vertical  line  passing  through 
the  center  of  the  pulley  :  What  is  the  relation  between  P  and 
W?  Ans.  P=W. 

3.  A  weight  of  240  Ibs.  is  sustained  by  a  power  equivalent  to 
7£  Ibs.  by  means  of  the  system  of  pulleys  described  in  Art.  123  : 
What  is  the  number  of  pulleys  ?  Ans.  5  pulleys. 

4.  What  power  will  be  necessary  to  sustain  a  weight  of  2387 
Ibs.  in  a  system  of  10  pulleys,  constructed  according  to  Fig.  88, 
where  the  strings  are  all  fastened  to  the  weight  1    Ans.  2^  Ibs 


CHAPTER  VIII. 

OF  THE  INCLINED  PLANE,  THE  SCREW,  AND  THE  WEDGE. 

128.  This  chapter  will  comprehend  the  three  remaining  me- 
chanical powers ;  viz.  the  Inclined  Plane,  the  Screw,  and  the 


NATURAL   PHILOSOPHY. 


Wedge;  beginning  with  the  Inclined  Plane,  being  that  upon 
which  the  properties  of  the  Screw  more  immediately  depend. 

THE   INCLINED   PLANE. 

129.  In  the  inclined  plane,  an  equilibrium  is  produced,  when  the 
power  is  to  the  weight,  as  the  sine  of  the  inclination* of  the  plane  is 
to  the  sine  of  the  angle,  which  the  direction  of  the  power  makes  with 
a  perpendicular  to  the  plane,  at  the  point  where  the  weight  rests 
upon  it. 

Let  AC  (Fig.  89,)  be  an  in- 
clined plane,  whose  length  is  g* 
AC,  height  AB,  and  base  BC ; 
and  suppose  the  weight  W  to 
be  kept  in  equilibrio  by  any 
other  weight  (or  power)  P  act- 
ing freely  over  a  pulley  fixed  at 
D.  Draw*  WE  at  right  angles 
to  AC,  meeting  AB  (produced  if 
necessary)  in  E  ;  the  weight  W 
may  be  considered  as  kept  at 
rest  by  three  forces,  viz.  the  ac- 
tion of  the  power  in  the  direc- 
tion WD,  its  own  weight  (or 
gravity)  in  direction  DE,  and  \|E 
the  reaction  of  the  plane  in  direction  EW  ;  /.  (by  Art.  56,)  these 
three  forces  are  to  each  other  as  the  three  sides  of  the  triangle 
DWE,  in  the  directions  of  which  they  respectively  act.  Hence, 
P  :  W  : :  WD  :  DE  : :  sin.  WED  or  ACB  :  sin.  DWE. 

By  the  third  law  of  motion,  the  pressure  of  W  upon  the  plane 
must  be  equal  to  the  reaction  of  the  plane  upon  W  ;  if,  therefore, 
EW  represents  that  reaction,  WE  will  represent  the  pressure 
upon  the  plane  ;  hence, 

P :  press,  on  the  plane : :  WD :  WE ::  sin.  WED  or  ACB :  sin.  WDE ; 
W :  press,  on  the  plane : :  DE :  WE ::  sin.  DWE  :  sin.  WDE. 

130.  In  the  inclined  plane,  when  the  power  acts  PARALLEL  TO  THE 
PLANE,  1.   The  power  is  to  the  weight  as  the  height  of  the  plane  to  its 
length  ;  2.  The  power  is  to  the  pressure  on  the  plane,  as  the  height 
of  the  plane  to  its  base ;  3.  The  weight  is  to  the  pressure  on  the 
plane  as  the  length  of  the  plane  to  its  base. 

If  the  power  acts  parallel  to  the  plane,  then  WD  may  be  con- 
sidered as  coinciding  with  WA,  and  the  power,  the  weight,  and 
the  pressure,  will  be  respectively  represented  by  the  three  sides 
WA,  AE,  WE,  of  the  triangle  AWE  ;  hence, 

*  This  figure  is  to  be  considered  as  a  section  of  the  plane  passing  through  the  cen- 
ter of  gravity  of  the  weight ;  and  if  the  weight  be  not  large,  that  center  of  gravity 
may  be  considered  as  placed  in  the  angular  point  (W)  of  the  triangle  DWE. 


MECHANICS.  109 

P  :  W  :  :  WA  :  AE  :  :  AB  :  AC  :  :  height  :  length. 

P  :  press,  on  the  plane  :  :  WA  :  WE  :  :  AB  :  BC  :  :  height  :  base. 

W  :  press,  on  the  plane  :  :  AE  :  WE  :  :  AC  :  BC  :  :  length  :  base. 

131.  In  the  inclined  plane,  whin  the  power  acts  parallel  to  the 
UASE  of  the  plane,  1.   The  power  is  to  the  weight,  as  the  height  of  the 
plane  to  its  base  ;  2.  The  power  is  to  the  pressure  on  the  plane  as 
the  Jieight  of  the  plane  to  its  length  :  3.   The  weight  is  to  the  pres- 
sure on  the  plane,  as  the  base  of  the  plane  to  its  length. 

If  the  power  acts  parallel  to  the  base  of  the  plane,  (i.  e.  if  the 
weight  W  be  sustained  upon  the  plane  by  a  force  acting  in  the 
direction  pW,  and  pushing  horizontally  against  the  plane,)  then 
produce  p  W  to  F  ;  and  when  there  is  an  equilibrium,  the  power, 
the  weight,  and  the  pressure  will  be  respectively  represented  by 
the  three  sides  WF,  FE,  WE,  of  the  triangle  WFE  ;  therefore, 
P  :  W  :  :  WF  :  FE  :  :  AB  :  BC  :  :  height  :  base. 
P  :  press,  on  the  plane  :  :  WF  :  WE  :  :  AB  :  AC  :  :  height  :  length. 
W  :  press,  on  the  plane  :  :  FE  :  WE  :  :  BC  :  AC  :  :  base  :  length. 

132.  The  least  power  will  be  required  to  raise  or  sustain  a  given 
weight  upon  a  given  inclined  plane,  when  the  direction  in  which  that 
power  acts  is  parallel  to  the  plane  ;  and,  conversely,  the  greatest 
weight  will  also  be  raised  or  sustained  by  a  given  power  upon  a 
given  inclined  plane,  when  the  direction  in  which  the  power  acts  is 
parallel  to  the  plane. 

Let  a—  angle  of  inclination  of  the  plane,  /3=angle  which  the 
direction  of  the  power  makes  with  a  perpendicular  to  the  plane 
at  the  point  where  the  weight  rests  upon  it;  then,  by  Art.  129, 


T>       W  r>  -n  .  _,  ._Tr 

P  :  W  :  :  sin.  a  :  sm.  /3  ;  .-.  P=  —  :  —  -.  —     Suppose  W  and  sin.  a 

to  be  given,  then  P  varies  as  -:  —  -g  and  will  consequently  be 

the  least  when  sin.  (3  is  the  greatest,  i.  e.  when  the  angle  DWE 
becomes  a  right  angle,  or  P  acts  parallel  to  the  plane. 

Again,  W=  —  :  —  -  —  ;  if,  therefore,  P  and  sin.  a  be  given,  then 
sin.  a 

W  oc  sin.  /3,  and  will  consequently  be  greatest  when  sin.  p  is 
greatest. 

133.  The  pressure  on  a  given  inclined  plane,  with  a  given  power, 
is  greatest  when  the  power  acts  parallel  to  the  base  of  the  plane. 
Let  7=WDE  ;  then,  by  Art.  129,  P  :  pressure  upon  the  plane 

:  :  sin.  a  :  sin.  y,  .'.pressure  upon  the  plane  =  —  :  --  —  ;  if  P  and 

sm.  a 

sin.  a  be  given,  then  the  pressure  upon  the  plane  oc  sin.  y,  and 
wi«ll  consequently  be  greatest  when  sin.  y  is  greatest,  i.  e.  when 


110 


NATURAL   PHILOSOPHY. 


WDE  becomes  a  right  angle,  or  the  power  acts  parallel  to  the 
base  of  the  plane.* 

134.  When  a  weight  W  is  sustained  by  another  weight  P  go- 
ing over  a  fixed  pulley  D,  (Fig.  90,)  since  the  angle  DWE  varies 
at  every  point  of  the  plane,  it  is  evident  that  there  is  but  one 
point  of  the  plane  where  a  given  power  will  sustain  a  given 
weight ;  and  that  point  may  be  thus  determined.  Take  BG  : 
BC  : :  P  :  W,  and  with  center  B  and  radius  BG,  describe  a  circu- 
lar arc  cutting  AC  in  F ;  join  BF,  and  draw  DKL  perpendicular 
to  BF,  or  to  BF  produced :  then  the  point  W,  where  DL  cuts 
D  Fig.  90.  » 


C      G   L 


AC,  will  be  the  point  required ;  for  the  triangles  DWE,  BFC  be- 
ing  similar,!  we  nave 

WD  :  DE  : :  BF  or  BG  :  BC ; 
butP:  W::BG  :  BC, 

/.P:  W::  WD  :  DE  ;  hence, 

by  Art.  129,  the  relation  between  the  sides  of  the  triangle  WDE 
is  such  as  to  give  the  position  W  when  there  is  an  equilibrium 
between  P  and  W. 

But  when  the  power  acts  parallel  to  the  plane  or  parallel  to 
the  base  of  the  plane,  the  ratio  of  P :  W  is  constant  J  through  every 
part  of  the  plane  ;  if  therefore  a  given  power  be  in  equilibrio 

*  In  this  case,  since  WE  is  greater  than  FE,  it  appears  that  the  pressure  is  greater 
than  W  ;  but  in  this  there  is  no  inconsistency,  for  it  is  evident  that  when  a  weight  is 
sustained  upon  an  inclined  plane  by  means  of  a  force  acting  in  direction  pW,  part  of 
the  pressure  arises  from  the  power  as  well  as  from  the  weight,  and  therefore  the  whole 
oressure  may  be  greater  than  the  weight. 

t  Draw  WE  at  right  angles  to  AC,  as  before,  then  AEW=ACB ;  and  in  the 
right-angled  triangles  BKL,  DEL,  KLB  is  common,  .-.  KBL=KDB ;  hence  CFB 
=  DWE,  and  consequently  the  triangle  CFB  is  similar  to  the  triangle  DWE. 

t  For  (in  one  case)  P  :  W  : :  height  of  the  plane  :  length  of  the  plane  ;  and  in 
the  other,  P  :  W  : :  height  of  plane  :  base  of  plane.  (Art.  130,  131.) 


MECHANICS.  Ill 

with  a  given  weight  at  any  one  point  of  a  plane,  the  same  power 
would  also  be  in  equilibrio  with  the  weight,  when  placed  at  any 
other  point  of  that  plane.  We  now  proceed  to  give  a  few  ex 
amples  for  illustration. 

135.  EXAMPLES. 

1.  A  person  is  just  able  to  sustain  by  his  strength  a  weight  of 
200  pounds.  What  weight  would  he  be  able  to  sustain  on  an 
inclined  plane  whose  elevation  is  50°,  by  means  of  a  rope  going 
round  it,  and  fixed  to  the  top  of  the  plane  in  the  manner  repre- 
sented in  the  annexed  figure  ? 

In  this    case,  (Fig.  91,)  the    power       Fig.  91. 
which  supports  the  weight  acts  parallel 
to  the  plane,  /.by  Art.  130,  P  :  W  : : 
AB  :  AC  : :  sin.  ACB  (50°)  :  radius  : :  /     a  A 


7660  :  10000,  or  W=;  this  is 


the  weight  supported  by  the  rope  PWA  ; 

but  since  that  rope  is  fixed  at  A,  each 

part  PW,  WA,  of  that  rope  supports     >fl|jf 

half*  the  weight  ;  hence,  if  the  force  jf  il1""1"1"""111"1111111  .......  "l|""""""i 


exerted  by  the  rope  PWA=P,  the  force 

.acting  in  the  direction  WP=£P  ;  calling  that  force  (p,}  thenp^P, 

or  P=2p  ;  substitute  this  for  P  and  we  have 

W=M«=(if  ^200)4-^=532.19  pounds; 

if  therefore  a  person  by  his  natural  strength  is  able  to  lift  a  weight 
of  200  pounds,  acting  under  the  circumstances  here  represented, 
he  will  be  able  to  sustain  a  weight  of  522.19  pounds. 

2.  Upon  an  inclined  plane,  whose  length  is  20  feet,  and  eleva- 
tion 30°,  a  weight  of  3  pounds  is  sustained  by  a  power  of  2 
pounds,  acting  over  a  pulley  fixed  at  the  distance  of  10  feet  from 
the  top  of  the  plane,  (in  the  manner  represented  in  Fig.  90  :)  It 
is  required  to  find  the  distance  of  W  from  the  top  of  the  plane, 
when  there  is  an  equilibrium. 

Since  AC=20  feet,  and  ACB=30°,  AB  (=sin.  30°)  will  be 
equal  to  10  feet,  and  consequently  BC=v/(AC2-BA2)=\/300 
=17.32  feet  ;  constructing  the  figure,  therefore,  as  in  Art.  134, 

2x17  S2 

we  have  BG  :  BC(17.32)  :  :  P(2)  :  W  (3),  .-.BG  or  BF=  -  -— 

o 

=11.55  feet;  hence,  in  the  triangle  BFC  we  have  BC=17.32, 

*  The  case  being  similar  to  that  of  a  weight  supported  by  two  parallel  strings  go- 
ing  over  a  pulley. 


112  NATURAL   PHILOSOPHY. 

BF=11.55,  FCB=30°,  from  which  the  angles  FBC,  CFB  are 
found  to  be  respectively  18°  34'  and  131°  26' ;  but  the  triangle 
DWE  is  similar  to  the  triangle  BFC,  /.WED=30°,  WDE=18° 
34',  and  DWE=131°  26'. 

Again,  since  DWE  is  131°  26',  and  AWE  a  right  angle,  DWA 
must  be  41°  26' ;  hence,  in  the  triangle  DWA,  we  have  AD=10 
feet,  WDA=18°  34',  and  DWA=41°  26',  from  which  AW  is 
found  to  be  4.812  feet,  which  gives  the  distance  of  W  from  the 
top  of  the  plane  when  P  is  in  equilibrio  with  W. 

3.  A  body  is  sustained  upon  an  inclined  plane,  first  by  a  power 
acting  parallel  to  the  plane,  and  afterward  by  a  power  acting 
parallel  to  the  base  of  the  plane.     Compare  the  pressures  upon 
the  plane  in  these  two  different  cases. 

By  Art.  130,  when  the  power  acts  parallel  to  the  plane, 
W  :  press,  on  the  plane  (P)  : :  length  :  base. 

By  Art.  131,  when  the  power  acts  parallel  to  the  base, 
Press,  on  the  plane  (p)  :  W : :  length  (L)  :  base  (B  ;)  /. 
» :  P : :  L2 :  B2. 

Thus,  in  a  plane  whose  elevation  is  60°,  (and  whose  length  is 
consequently  double*  of  its  base,)  it  makes  a  difference  of  4  :  1 
as  to  the  pressure  upon  the  plane,  whether  a  body  is  sustained 
upon  it  by  a  force  acting  parallel  to  the  plane,  or  by  one  acting 
parallel  to  the  base  of  the  plane. 

4.  Two  weights  P,  W,  resting  upon  the  inclined  planes  AC, 
AD,  (Fig.  92,)  whose  common  height  is  AB,  keep  each  other  in 
equilibrio  by  means  of  a  string  going  over  a  pulley  fixed  at  A. 
Compare  the  two  weights. 

Since  the  string  passes  freely  over 
the  pulley  at  A,  and  the  two  weights 
are  at  rest,  it  is  evident  that  the  ten- 
sion of  the  string  WAP  must  be  every- 
where the  same,  i.  e.  whatever  power 
is  exerted  at  A  to  sustain  W  on  the 
plane  AC,  the  same  is  exerted  at  that 
point  to  sustain  P  upon  the  plane  AD  ;  call  that  power  (p,)  then, 
since  the  power  in  each  case  may  be  considered  as  acting  pa- 
rallel to  the  plane,  we  have,  by  Art.  130. 

p  :  W  : :  AB  :  AC,  and  P  :  p : :  AD :  AB ; 

.*.  P  :  W  : :  AD :  AC  : :  plane  upon  which  P  rests  :  plane  upon 
which  W  rests.  Hence,  two  weights  resting  on  two  inclined 
planes  which  meet,  (forming  a  ridge.)  will  balance  each  other, 
when  they  are  to  one  another  as  the  lengths  of  the  planes  on 
which  they  respectively  rest. 

*  In  Fig.  91,  if  ACB=60°,  then  CAB=30° ;  .-.  BC=i  radius=lAC,  or  AC=2BC 


MECHANICS.  113 

5.  A  body  is  supported  between  two  inclined  planes  of  g^iven 
elevations  :  Compare  the  pressure  upon  the  planes. 

Let  NHL,  (Fig.  93,)  represent 
a  perpendicular  section  of  the 
body  passing  through  its  two 
points  of  contact  H,  L,  with  the 
planes ;  and  from  those  points 
draw  HF,  LF  at  right  angles  to 
the  planes  DC,  AC  respectively. 
From  their  intersection  F,  draw 
FG  perpendicular  to  the  horizon, 
and  let  it  represent  the  weight  of 
the  body.  Through  G  draw  GO  E  1 
parallel  to  LF ;  then  the  three 
sides  of  the  triangle  GOF  will  be  in  the  direction  of  the  three 
forces  which  keep  the  body  at  rest  upon  the  plane  AC,  viz.  GO 
will  represent  the  reaction  of  the  plane  AC  ;  OF  the  reaction  of 
the  plane  DC  ;  and  FG  the  weight  of  the  body  ;  and  in  the  same 
manner  it  may  be  shown  that  the  three  sides  of  the  same  triangle 
will  represent  the  three  forces  which  keep  the  body  at  rest  upon 
the  plane  DC.  Through  G  draw  MGK  parallel  to  the  horizon ; 
then  since  the  three  sides  of  the  triangle  GOF  are  perpendicular 
to  the  three  sides  of  the  triangle  MCK,  MCK  must  be  similar 
to  GOF,  (see  note,  p.  55  ;)  and  since  the  weight  of  the  body,  the 
pressure  upon  the  plane  DC,  and  the  pressure  upon  the  plane 
AC,  are  respectively  represented  by  the  three  sides  FG,  FO,  OG 
of  the  triangle  GOF,  they  will  also  be  represented  by  the  three 
sides  MK,  MC,  CK,  of  the  triangle  MCK.  Hence, 

Pr.  on  DC  :  Pr.  on  AC  : :  MC  :  CK 

: :  sin.  MKC  or  ACB  :  sin.  CMK  or  DCE. 

.-.  Pr.  on  DC  :  Pr.  on  AC  : :  sin.  ACB  :  sin.  DCE. 

Thus  suppose  DCE  =60°,  ACB=30°,  then  pressure  on   DC: 

/O 

pressure  on  AC  : :  sin.  30°  :  sin.  60°  : :  |  :  — - : :  I  :  \/3.     Hence, 

when  a  weight  is  supported  between  fwo  inclined  planes,  the 
pressures  on  the  planes  are  reciprocally  as  the  sines  of  the  angles 
of  inclination  of  the  planes. 


THE  SCREW. 

136.  The  screw  is  a  spiral  thread  or  groove,  winding  round  a 
cylinder,  so  as  to  cut  all  the  lines  drawn  on  its  surface  parallel  to 
its  axis,  at  the  same  angle.  The  spiral  may  be  either  on  the  convex 
or  the  concave  surface  of  the  cylinder,  and  the  screw  is  denominated 
accordingly,  the  external  or  the  internal  screw. 

The  distance  between  the  two  contiguous  threads  of  a  screw. 
15 


114 


NATURAL   PHILOSOPHY. 


corresponds  to  the  height  of  an  inclined  plane,  and  the  circum 
ference  of  the  cylinder  corresponds  to  the  base  of  the  same 
plane  ;  hence  the  forces  necessary  to  produce  an  equilibrium  in 
the  screw,  are  the  same  as  in  the  inclined  plane.  Thus  let  the 
inclined  plane  ABC  (Fig.  94)  be  wrapped  round  a  cylinder,  the 

Fig.  94. 


A  B 

circumference  of  whose  base  is  equal  to  the  line  AB ;  then  the 
point  A  being  pl-aced  on  A',  the  point  B  will  come  round  to  A', 
and  the  point  C  will  fall  on  C',  and  the  line  AC  will  trace  out 
the  thread  of  the  screw  on  the  surface  of  the  cylinder  as  far  as 
C',  and  maybe  continued  in  the  same  manner.  By  Art.  131, 
when  the  power  acts  parallel  to  the  base  of  the  plane,  an  equi- 
librium is  produced  when  the  power  is  to  the  weight  as  the 
height  of  the  plane  to  its  base  ;  or,  applied  to  the  screw,  an  equi- 
librium is  produced,  when  the  power  is  to  the  weight,  as  the 
distance  between  two  contiguous  threads  is  to  the  circumference 
of  the  base. 


137.  Let  the  external  and  in-  Fig.  95 

ternal  screws  be  fitted  to  each 
other  in  the  manner  represented 
in  Fig.  95,  and  let  the  external 
screw  be  turned  round  by  a  power 
applied  to  the  lever  BC,  (acting 
parallel  to  the  base  of  the  cylin- 
der,) while  the  internal  screw  re- 
mains fixed ;  then  it  is  evident, 
from  the  manner  in  which  the 
two  screws  act  upon  each  other, 
that  while  the  lever  BC  makes 
one  revolution,  the  external  screw 
will  be  elevated  or  depressed 
through  one  of  the  spaces  db,  be, 
according  to  the  direction  in 
which  it  is  turned.  When  the 
screw  is  depressed  it  drives  be- 
fore it  the  board  EF,  which  moves 
in  the  grooves  DE,  NF,  by  which  means  a  pressure  is  created 


MECHANICS.  115 

upon  any  substance  (S)  placed  between  that  board  and  the  fixed 
board  KM. 

138.  Let  us  now  estimate  the  quantity  of  this  pressure,  by 
finding  the  relation  which  it  bears  to  the  power  which  produces 
it.     To  do  this,  it  will  be  necessary,  in  the  first  place,  to  consider 
the  force  which  would  be  generated  in  the  elevation  of  the  screw; 
which  force  may  be  estimated  by  showing  separately  what  part 
of  it  arises  from  the  action  of  the  spirals  of  the  screw  upon  each 
other,  and  what  from  the  action  of  the  lever.     As  the  machine 
turns  round,  each  point  of  the  external  screw  acts  upon  the  cor- 
responding one  of  the  internal  screw,  with  a  force  analogous  to 
that  by  which  a  body  is  sustained  upon  an  inclined  plane  when 
the  power  acts  parallel  to  the  base  of  the  plane  ;*  the  whole 
force  therefore  of  the  screw  will  be  of  the  same  kind,  and  (by 
Art.  131)  will  bear  to  the  weight  which  it  could  support,  the  ratio 
of  the  distance  between  two  spirals  to  the  circumference  of  the 
cylinder.     This  would  be  the  case,  supposing  the  force  to  act 
close  to  the  surface  of  the  cylinder  ;  when  it  acts  therefore  at 
the  extremity  of  the  lever  BC,  it  will  be  increased  in  the  pro- 
portion of  the  length  of  the  lever  to  the  radius  of  the  cylinder. 

139.  In  the  screiv,  an  equilibrium  is  produced  when  the  power  is 
to  the  weight,  as  the  distance  between  two  contiguous  threads  is  to 
the  circumference  of  tfie  circle  described  by  one  revolution  of  the 
power. 

Let  <Z=the  distance  between  two  Fi&-  96- 

spirals  of  the  screw,  which  is  cut 
upon  the.  cylinder  of  which  AFG 
(Fig.  96)  is  the  section,  a=length 
of  the  lever  (CB),  *=3.1415,  &c., 
then  the  circumference  (BDE)  of 
the  circle  described  by  the  extremi- 
ty of  the  lever  =  2-ra.  Let  P  =  the 
power  acting  at  the  extremity  of 
the  lever  ;  p  =  the  power  acting  at 
the  surface  of  the  cylinder,  W=the 
weight  which  is  kept  in  equilibrio 
by  P,  then,  by  Art.  138, 

p  :  W  :  :  d  :  circumference  AFG, 

P  :p  :  :  CA  :  CB  :  ;  circumf.  AFG  ;  circumf.  BDE  ; 

.-.  P  :  W  :  :  d  :  circumf. 


Wd  w    2*«P  J     2     i>        ,       Wd 
—  ,  W=-^-,  d=-w-,  and  a=^ 


*  Instead  of  moving  a  body  up  an  inclined  plane,  we  here  move  the  plane  itself 
against  a  resistance,  which  is  overcome  in  the  same  manner  as  th«.t  of  a  body,  and 
which  may  therefore  be  properly  c<  nsidered  as  a  weight. 


116  NATURAL   PHILOSOPHY. 

hence,  if  any  three  of  the  four  quantities  W,  P,  a,  d,  be  given, 
the  fourth  may  be  found.* 

140.  We  have  thus  estimated  the  magnitude  of  the  weight 
which  might  be  sustained  by  a  given  power  applied  to  the  ele- 
vation of  the  screw  ;  but  this  machine  is  oftener  used  for  the 
purpose  of  creating  a  pressure  than  for  raising  a  weight  ;  and  it 
is  evident,  that  whatever  force  is  exerted  by  the  screw  to  sus- 
tain a  weight  when  it  is  turned  in  one  direction,  will  also  be  ex- 
erted to  create  a  pressure  downward,  when  it  is  turned  in  an 
opposite  direction.  Whether,  therefore,  the  screw  be  applied  to 
raising  a  weight,  or  creating  a  pressure,  the  power  necessary  to 
sustain  the  weight  or  produce  the  pressure,  will  always  bear  to 
that  weight  or  pressure  the  ratio  of  the  distance  between  any 
two  spirals  of  the  screw,  to  the  circumference  of  the  circle 
which  the  power  describes. 

14.1.  EXAMPLES. 

1.  A  screw,  the  distance  between  whose  spirals  is  one  inch,  is 
turned  horizontally  by  a  lever  whose  length  is  2  feet,  reckoning 
from  the  axis  of  the  screw  :  What  weight  could  be  sustained  or 
pressure  produced  by  it,  when  a  power  of  30  pounds  acts  at  the 
extremity  of  the  lever  ? 

r    2«aP    2x3.1415x24x30 
By  Art.  139,  W=—  j~=—       —  J^        -  =4523.76  pounds  ; 

i.  e.  a  power  of  30  pounds  applied  to  a  machine  of  this  kind 
would  be  sufficient  to  sustain  a  weight,  or  create  a  pressure, 
equivalent  to  about  4523  pounds,  or  somewhat  more  than  two 
tons. 

2.  A  person  who  could  just  lift  a  weight  of  60  pounds  found 
himself  able,  by  means  of  a  lever  3  feet  long,  acting  as  a  handle 
to  a  screw,  to  sustain  a  ton  weight  :   What  was  the  distance 
between  the  spirals  of  the  screw  ? 

2-raP    2X3.1415X3X60 
By  Art.  139,  d=^-=  --        ---  =>5°     eet  =  about  6 


inches. 

3.  In  Fig.  97,  the  screw  AB,  which  is  turned  by  a  power  P 
acting  upon  the  handle  PQ,  turns  at  the  same  time  the  wheel  C, 

*  From  this  proportion  it  appears  that  the  relation  P  :  W  depends  entirely  upon 
the  distance  between  the  spirals  and  the  circumference  which  the  power  describes, 
whatever  be  the  thickness  of  the  cylinder  upon  which  the  screw  is  cut  :  and  since 

P  l=—  —  I  oc-  —  ,  when  W  is  given,  PCX—;  i.  e.  the  power  necessary  to  sustain  a 

given  weight  varies  directly  as  the  distance  between  the  spirals,  and  inversely  as 
the  length  of  the  leve- 


MECHANICS.  117 

in  such  a  manner  as  to  cause  it  to  draw  up  the  weight  W,  by  a 
rope  wound  round  the  axle  D  :  This  is  called  the  endless  screw, 
and  it  is  required  to  find  the  ratio  of  P  :  W. 

Supposing  all  the  parts  of  this  machine  to  be  nicely  adjusted, 
it  is  a  very  powerful  one  ;  inasmuch  as  it  combines  the  energy 
of  the  screw  with  the  multiplying  power  of  the  wheel  and  axle. 

Fig.  97. 


To  estimate  the  effect,  therefore,  let  PQ=«,  the  distance  between 
two  spirals  of  the  screwed,  radius  of  the  wheel=R,  radius  of 
the  axle=r,  #=3.14159,  and  Q=the  force  exerted  by  the  screw 
upon  the  wheel ;  then,  by  Art.  139,  P  :  Q, : :  d :  2*«, 

~.:W::r:R, 


and,  by  Art.  113, 


.'.P:W::dr:2«aR. 

__          _     Wdr        .  ___    2P*aR     .        .    ,    .     .  ,     , 

Hence  P  =-  —  =;.  and  W=  —  ,  -  ;  let  d—\  inch,  a=12  inches, 
ar 


r=4  inches,  R=18  inches,  and  P=30  pounds,  then  W=  — 

pounds  =4.54  tons  ;  so  that,  by  means  of  this  machine,  a  power 
of  30  pounds  would  be  sufficient  to  keep  in  equilibrio  a  weight 
of  about  4|  tons. 

142.  QUESTIONS  ON  THE  PRINCIPLES  OP  THE   INCLINED   PLANE  AND 
SCREW. 

1.  If  a  man  can  draw  a  weight  of  125  pounds  up  the  side  of 
a  perpendicular  wall,  20  feet  high,  what  weight  will  he  be  able 
to  raise  along  a  smooth  plank  44  feet  long,  laid  sloping  from  the 
top  of  the  wall  1  Arts.  275  pounds. 

2.  Suppose  that  a  horse  is  able  to  draw  a  weight  of  440 
pounds  out  of  a  well,  (by  a  rope  passing  over  a  fixed  pulley, 
which  allows  the  horse  to  draw  in  a  horizontal  direction  ;)  what 


118 


NATURAL    PHILOSOPHY. 


weight  will  the  same  animal  draw  up  a  railway  having  a  slope 
of  five  degrees,  no  allowance  being  made  for  friction  ? 

Ans.  5048.5  pounds. 

3.  A  lever  five  feet  long  is  fixed  at  right  angles  in  a  screw 
whose  threads  are  one  inch  asunder,  so  that  the  lever  turns  just 
once  round  in  raising  or  depressing  the  screw  one  inch.     If  then 
this  lever  be  urged  by  a  force  of  65  pounds,  with  what  force 
will  the  screw  press  ?  Ans.  24504.35  pounds. 

4.  A  shipwright  wishing  to  haul  a  ship  upon  the  stocks,  em- 
ployed a  machine,  combining  the  lever,  the  screw,  the  wheel 
and  axle,  the  pulley,  and  the  inclined  plane,  as  represented  in 
Fig.  98. 


Fig.  98. 


"*"*  e  handle  of  the  winch  BC  =18  inches. 

The  distance  of  the  threads  on  CD=1  inch.     , 
The  radius  of  the  wheel  ED  =2  feet. 

The  radius  of  the  axle  EF  =6  inches. 

G  is  a  fixed,  and  H  a  movable  pul- 
ley, the  number  of  strings  being  =4. 
Inclination  of  the  plane  =30°. 

Allowing  a  man  to  turn  on  the  handle  at  B,  with  a  force  equal 
to  100  pounds,  how  much  force  could  he  exert  on  the  ship  ? 

Ans.  361911.168  pounds,  or  more  than  161$  tons. 


THE    WEDGE. 

143.  All  those  instruments  which  are  used  for  the  separation 
of  the  parts  of  bodies,  such  as  knives,  axes,  coulters,  and  chisels, 
come  under  the  general  denomination  of  the  wedge  ;  but  these 
instruments  are  made  of  such  variety  of  shapes,  and  forces  are 
applied  to  them  in  such  various  ways,  that  of  all  the  mechanical 
powers,  the  wedge  is  that  whose  properties  are  least  capable  of 
being  brought  to  mathematical  calculation.  In  the  particular 
case  where  the  wedge  is  of  the  form  of  a  triangular  prism,  and 
the  resistance  upon  its  sides  can  be  considered  as  forces  acting 
in  given  directions,  the  relation  between  those  resistances  and 


MECHANICS. 


119 


the  power  which  counteracts  them,  may  be  estimated  in  the  fol- 
lowing manner. 

144.  In  the  wedge,  an  equilibrium  is  produced  when  that  part  of 
the  power,  which,  when  resolved,  acts  perpendicularly  to  the  bach  of 
the  wedge,  is  equal  to  the  sum  of  those  parts  of  the  resistances  which 
also  act  perpendicularly  to  the  back  ;  these  resistances  being  to  each 
other  inversely  as  their  respective  distances  from  the  line  of  direc- 
tion in  which  the  resultant  of  the  power  acts. 

Let  ABC  (Fig.  99)  represent  a  FiS-  "• 

section  of  the  wedge  perpendicular 
to  the  axis  of  the  prism,  and  sup- 
pose its  sides  AC,  BC,  to  be  per- 
fectly smooth.  Let  a  power  P, 
whose  magnitude  and  direction  is 
represented  by  ab,  act  upon  AB, 
the  back  of  the  wedge,  and  let  it 
be  counteracted  by  two  resistances 
R,  R',  (which  are  represented  in 
quantity  and  direction  by  the  lines 
de,  U,)  acting  upon  the  sides  AC,  BC.  Resolve  ab  into  ac  per- 
pendicular, and  be  parallel  to  the  back  of  the  wedge,  and  let  de 
also  be  resolved  into  df  perpendicular  and  ef  parallel  to  the  side 
AC  ;  and  since  the  side  AC  is  perfectly  smooth,  df  only  is  effec- 
tual to  stop  the  progress  of  the  wedge  ;  resolve  df  again  into  dg 
parallel,  and  gf  perpendicular  to  the  back  of  the  wedge,  then  fg 
is  the  only  part  of  the  resistance  R  which  is  directly  opposed  to 
that  part  of  the  ppwer  (viz.  ac}  which  acts  perpendicularly  to 
the  back  of  the  wedge.  Let  the  resistance  R'  be  resolved  in  the 
same  manner,  and  let  mn  be  that  part  of  it  which  is  directly  op- 
posed to  ac  ;  then,  supposing  every  part  of  the  wedge  to  be  per- 
fectly hard  and  inflexible,  it  is  evident  that  in  the  case  of  an 
equilibrium  between  P  and  R+R',  ac=fg-\-mn. 

But  in  ascertaining  this  relation  between  P  and  R+R'  when 
they  are  in  equilibrio,  it  should  be  recollected  that  the  point  c. 
cannot  be  arbitrarily  assumed  ;  for  there  evidently  will  be  a 


tendency  to  vibratory  or  rotary  motion, 
unless  the  two  resistances  balance 
themselves  about  that  point.  To  de- 
termine the  point  c  so  that  this  ten- 
dency to  vibratory  motion  shall  be 
prevented,  produce  gf,  nm,  (Fig.  100,) 
to  k  and  o,  then  it  is  evident  that  the 
resistances  fg,  mn  will  only  balance 
themselves  about  c,  when,  according 
to  Theorem  II,  of  parallel  motion, 
(Art.  60,)  fg  :  mn  :  :  co  :  ck  ;  that  is, 
when  the  effective  parts  of  the  resist- 


Fig.  100. 


rZO  NATURAL  PHILOSOPHY. 

ances  are  to  each  other  inversely  as  their  respective  distances 
from  the  line  of  direction  in  which  the  resultant  of  the  power 
acts. 

145.  But  for  the  purpose  of  computing  the  actual  power  of  the 
wedge  in  particular  cases,  we  must  find  an  expression  for  the 
power  in  known  terms,  arising  from  the  shape  of  the  wedge,  and 
the  conditions  of  the  resistances.  Therefore  produce  dg  to  k, 
(Fig.  99,)  and  let  a  perpendicular  CD,  be  drawn  from  the  verti- 
cal angle  to  the  back  of  the  wedge.  Let  a,  /3  denote  the  two 
vertical  angles,  *  the  angle  made  by  the  power  with  the  back 
of  the  wedge,  and  £  and  g  the  respective  angles  made  by  the  re- 
sistances R,  R'  with  the  sides.  Then,  since  in  the  right-angled 
triangles  dfh,  CAD,  the  alternate  angles  dhf,  CAD,  are  equal, 
the  angle  fdg  must  be  equal  to  the  angle  ACD  (a.)  Now 
de(R)  :  df  :  :  rad.  :  sin.  def,  or  sin.  £, 
and  df:fg  :  :  rad.  :  sin.  fdg,  or  sin.  a, 

_  —  —  i  Rxsin.  fxsin.  a 

.'.  R  :fg  :  :  rad.  I2  :  sm.  ^xsin.  a,  orfg=  -  =^  -  . 

rad.  I 

For  the  same  reason,  mn=  R/xslnl£/Xsin^  . 

rad.  |2 

Again,  a&(P)  :  ac  :  :  rad.  :  sin.  if,  .*.  ac=  -  —  . 

rad. 

But  when  there  is  an  equilibrium,  ac=fg-\-mn  ;  "hence 

Pxsin.  «    R  x  sin,  g  x  sin.  a+R'  x  sin.  g'  x  sin.  /?, 
~  " 


or  p_R  xsin.  f  xsin.  a+R'  x  sin.  g  xsin.  /S 

rad.  xsin.  -ir 

This  formula  enables  us  to  compute  the  power  of  a  wedge, 
(or  its  ratio  to  the  weight,)  when  we  have  given  the  angles  made 
by  a  perpendicular  drawn  from  the  vertical  angle  to  the  back 
of  the  wedge,  and  likewise  the  angles  at  which  the  power  and 
resistances  respectively  act,  whatever  these  angles  may  be  :  but 
when  (as  is  frequently  the  case)  the  power  acts  perpendicularly 
to  the  back,  and  the  resistances  perpendicularly  to  the  sides,  the 
formula  becomes  much  simpler.  For  then,  sin.  it,  sin.  f,  and 
sin.  £',  each  becomes  equal  to  radius,  and  the  general  formula 
becomes  P=Rxsin.  a+R'xsin.  /3. 

146.  In  the  particular  case  when  the  directions  of  the  power 
and  resistances  meet  in  the  same  point,  as  in  Fig.  101,  the  power 
is  to  the  weight  as  the  back  of  the  wedge  to  the  sum  of  the  sides  ; 
for  then  it  is  evident,  that  this  equilibrium  is  produced  under 
the  same  circumstances  as  that  of  a  body  kept  at  rest  by  three 
forces  whose  directions  meet  in  that  point;  but  (by  Art.  57) 


MECHANICS. 


121 


B 


those   three   forces  would  be   to  each  F'g- 101- 

other  as  the  three  sides  of  a  triangle 
perpendicular  to  their  respective  direc- 
tions ;  P,  R  and  R',  will  therefore  be  to 
each  other  as  the  three  sides  of  the  tri- 
angle ABC ;  i.  e. 

P:R::AB:AC;  P:R'::AB:BC 

.-.  P :  R+R' : :  AB :  AC+BC.* 

If  the   wedge  is   isosceles,   and  two 
equal  resistances  act  perpendicularly  to 
the  sides,  while  the  power  acts  perpen- 
dicularly to  the  back,  then  P :  W  : :  iAB  :  AC  ; 
For,  P  :  (2R)  W : :  AB  :  2AC 

.-.  P :  W : :  iAB  :  AC  ;  that  is, 

If  the  wedge  is  isosceles,  the  power  is  to  the  weight  as  half  the 
back  of  the  wedge  is  to  one  of  its  sides. 

Hence,  the  thinner  the  wedge,  that  is,  the  longer  the  sides  of 
the  wedge  in  proportion  to  its  thickness,  the  greater  is  its  power 
of  overcoming  resistance,  with  a  given  blow;  a  well  known 
property  of  sharp  instruments,  which  are  referred  to  the  wedge. 

147.  EXAMPLES. 

1.  A  wedge,  whose  sides  are  perfectly  polished  planes,  is 
driven  into  the  trunk  of  a  tree,  until  that  part  of  the  pressure  on 
each  side,  which  acts  perpendicularly  to  the  back,  is  1500  pounds : 
What  force  must  be  applied  at  the  back  to  prevent  its  recoil  ? 

Here  P=R+R'=2R=3000  pounds. 

2.  Two  equal  resistances  (R,  R')  acting  at  angles  of  60°  and 
30°  upon  the  sides  of  a  perfectly  smooth  wedge,  are  kept  in  equi- 
librio  by  a  power  acting  perpendicularly  to  its  back  ;  the  angle 
which  a  perpendicular  (from  the  vertical  angle  of  the  wedge  to 
the  back)  makes  with  the  side  upon  which  R  acts  is  45°,  and  with 
that  upon  which  R'  acts  30°  :  Required  the  ratio  of  P :  R+R' 
or2R. 

This  is  a  case  of  Art.  145,  where  R'=R,  sin.  *=rad.  =  l, 
.*.  P=R  (sin.  £  xsin.  a+sin.  f'  xsin.  /3  ;)  but, 

sin.  e=sin.  60°=— 


sin.  a=sin.  45C 

sin.  £'=sin.  30°= 
sin.  |8=sin.30°= 


2 


;  hence 
P:R::\/6+l:4,andP:2R 


: 8. 


*  Algebra,  Art.  388. 

16 


122 


NATURAL    PHILOSOPHY. 


148.  QUESTIONS  ON  THE  PRINCIPLES  OF  THE  WEDGE. 


1.  An  isosceles  wedge,  whose  acute  angle  was  5°,  was  in- 
serted in  a  cleft  of  a  rock.     The  pressure  exerted  perpendicu- 
larly on  each  side  was  equal  to  6500  pounds :  What  force  ap- 
plied at  the  back  of  the  wedge  would  overcome  the  resistance  ? 

Ans.  567.33  pounds. 

2.  In  a  wedge  of  the  form  of  a  triangular  prism,  the  power 
and  resistances  act  perpendicularly  to  the  back.     The  pressures 
upon  the  two  sides  of  the  wedge,  \vhose  lengths  are  5  and  7 
inches,  are  estimated  at  950  and  300  pounds  respectively,  acting 
at  the  center  of  each  side.     The  length  of  the  back  of  the  wedge 
is  5  inches  :  What  power  must  be  exerted  perpendicularly  to  the 
back  of  the  wedge,  to  produce  an  equilibrium  ? 

Ans.  P=1250  pounds. 

3.  A  force  of  1600  pounds,  acting  at  an  angle  of  75°  upon  the 
back  of  a  wedge,  is  just  sufficient  to  balance  two  resistances, 
which  are  to  one  another  as  5  :  7,  and  \vhose  directions  make  an- 
gles with  their  respective  sides  of  80°  and  40°.     The  angle 
which  a  perpendicular  from  the  vertical  angle  upon  the  back, 
makes  with  the  side,  upon  which  is  exerted  the  greater  resistance, 
is  10°,  and  that  which  it  makes  with  the  other  side  is  4°  :  What 
is  the  amount  of  the  resistances?  Ans.  1 6487. 76 pounds. 

GENERAL  PRINCIPLE,  APPLICABLE  TO  ALL  THE  MECHANICAL  POWERS. 

149.  The  most  simple  view  which  can  be  taken  of  the  me- 
chanical powers,  is  by  a  comparison  of  the  respective  VELOCITIES 
OF  THE  POWER  AND  THE  WEIGHT.  In  order  clearly  to  understand 
this  subject,  it  must  be  recollected, 

That  an  equilibrium  implies  the  action  of  opposite  and  equal 
FORCES  ; 

That  the  measure  of  a  force  is  its  MOMENTUM,  and,  consequently, 
that  in  an  equilibrium,  the  momenta  on  the  opposite  sides  are  equal ; 

That  momentum  is  compounded  of  the  quantity  of  matter  and 
velocity ;  and  hence,  that  a  small  body  may  have  as  much  mo- 
mentum as  a  large  one,  if  it  moves  over  as  much  greater  space  in 
the  same  time,  as  its  quantity  of  matter  is  less. 

Now  let  us  apply  the  foregoing  principles  to  the  mechanical 
powers. 

When  the  power  and  weight  are  in  equilibrio,  one  has  just  as 
much  momentum  as  the  other  ;  and  therefore  the  product  of  the 
weight  into  its  velocity,  equals  the  product  of  the  power  into  its 
velocity  ;  and  it  will  be  seen  by  reviewing  the  several  mechani- 
cal powers,  that  in  the  theorem  by  which  the  law  of  equilibrium 
is  in  each  case  enunciated,  the  line  into  which  the  power  or  the 
weight  is  multiplied,  is  proportional  to  the  space  over  which  it 


MECHANICS.  123 

moves  in  a  given  time,  and  therefore  (by  Art.  12)  is  the  measure 
of  its  velocity.  This  doctrine  will  be  clearly  comprehended  by 
reviewing  each  of  the  mechanical  powers  separately.* 

150.  Suppose  P  and  W  (Fig.  Fig- 101/- 

101',)  to  vibrate  in  equilibrium  on  p®- 
the  end  of  a  straight  lever,  PCW : 
they  will  describe  similar  arcs 
P/?,  Ww,  which  will  be  the  mea- 
sures of  their  respective  veloci- 
ties ;  or,  V  :  v  : :  Pp  :  Ww  : :  PC  : 
WC  : :  W  :  P,  .-.  V  :  v  : :  W  :  P. 

In  the  wheel  and  axle,  while  the  power  descends  through  a 
space  equal  to  the  circumference  of  the  wheel,  the  weight  ascends 
through  a  space  equal  to  the  circumference  of  the  axle,  .'.  the 
velocity  of  the  power  :  the  velocity  of  the  weight  : :  the  circum- 
ference of  the  wheel  :  the  circumference  of  the  axle  : :  radius  of 
the  wheel  :  radius  of  the  axle  : :  W  :  P. 

In  the  single  fixed  pulley,  the  power  and  weight  move  through 
equal  spaces  in  the  same  time,  /.the  velocity  of  the  power  is 
equal  to  the  velocity  of  the  weight ;  and  in  this  machine  P=W, 
/.velocity  of  power  :  velocity  of  weight  : :  W  :  P. 

In  the  system  of  n  pulleys,  (Fig.  85,)  while  the  power  descends 
through  any  space  (x,)  each  of  the  strings  belonging  to  the  block 
of  pulleys  to  which  the  weight  is  appended  is  shortened  by  -^x,.'. 
the  weight  ascends  through  a  space  equal  to  -{x  in  the  same  time 
that  P  descends  through  the  space  x  ;  hence  the  velocity  of  the 
power  :  the  velocity  of  the  weight : :  x  :  ~  : :  n  :  1  : :  W  :  P. 

In  the  system  of  pulleys  (Fig.  86,)  while  P  descends  through 
any  space  (x,)  the  pulley  F  is  raised  through  a  space —\x ;  the 
pulley  D  through  a  space =£x  ?x=±x ;  the  pulley  B  through  a 
space=ix  £#=!#;  hence  (if  w=the  number  of  pulleys)  velocity 
of  P :  velocity  of  W  : :  x  :  (i)£z  : :  2" :  1  : :  W  :  P. 

Let  ABC  (Fig.  102,)  be  an  inclined  plane,       Fig  102- 
up  which  the  weight  W  is  drawn  by  a  pow- 
er P  acting  over  a  pulley  at  A,  then  P  :  W 
: :  AB :  AC.     Draw  Wb  parallel  to  CB,  and 
ab  parallel  to  AB  ;  then  while  P  descends 
through  a  space  equal  to  Wa,  W  ascends 
upon  the  plane  through  the  same  space  ;  but  _ 
the  space  actually  described  by  W  in  this c 
time  in  the  direction  of  gravity,  is  ba  ;  .•.  (when  the  velocity  of 

*  In  estimating  the  increase  of  the  power  necessary  to  put  the  machine  in  motion, 
the  friction  of  its  different  parts  should  be  taken  into  the  account.  If  p=ihe  sum 
of  the  impediments  arising  from  friction,  and  P=the  power  which  would  kee{j  the 
weight  (W)  in  equilibrio,  then  it  is  evident,  that,  before  the  machine  can  be  put  in 
motion,  the  power  actually  employed  must  exceed  P+J>. 


124  NATURAL   PHILOSOPHY. 

the  power  and  weight  are  estimated  in  the  direction  in  which 
they  respectively  act) 

velocity  of  P  :  velocity  of  W : :  Wa  :  ab  : :  AC  :  AB  : :  W  :  P. 

In  the  screw,  while  P  describes  the  circumference  of  a  circle 
whose  radius  is  BC,  (Fig.  96,)  the  weight  is  elevated  or  depressed 
through  a  space  equal  to  the  distance  between  two  contiguous 
spirals  (d,)  .-.  the  velocity  of  P :  the  velocity  of  W : :  circumfe- 
rence (DBE,) :  d : :  W :  P. 

Finally,  in  the  wedge,  the  power  of  overcoming  the  resistance 
is  proportioned  (Art.  146)  to  the  acuteness  of  the  wedge  ;  and 
the  distance  to  which  the  parts  are  separated,  that  is,  the  space 
over  which  the  weight  moves,  when  compared  with  the  space 
through  which  the  power  (namely,  the  wedge  itself  in  the  direc- 
tion of  the  power)  moves,  is  constantly  diminished  as  the  acute- 
ness  of  the  wedge  is  increased. 

In  the  same  manner  we  might  trace  this  relation  between  the 
velocity  of  the  power  and  the  velocity  of  the  weight,  in  machines 
where  the  power  and  weight  act  obliquely  to  each  other  ;  but  as 
the  operation  then  becomes  somewhat  more  intricate,  and  as 
what  has  been  already  shown  is  sufficient  to  illustrate  the  truth 
of  our  proposition,  it  seems  unnecessary  to  pursue  the  investigation 
any  further. 

THE    ROPE    MACHINE. 

151.  If  a  body  fixed  to  two  or  more  ropes,  is  sustained  by  powers 
which  act  by  means  of  those  ropes,  this  assemblage  is  called  the  Fu- 
nicular or  Rope  Machine. 

152.  A  given  weight  is  in  equilibria  with  two  given  powers,  which 
are  equal  to  one  another,  and  which  pass  over  pulleys  situated  in 
the  same  horizontal  line,  when  either  power  is  to  the  weight  as  the 
sine  of  the  angle  formed  by  the  directions  of  the  power  and  the 
weight,  is  to  the  sine  of  the  angle  formed  by  the  directions  of  the 
two  powers. 

Let  A  and  B  (Fig.  103,)  be 
two  pulleys  fixed  at  a  given  dis- 
tance from  each  other  in  the 
same  horizontal  line,  and  sup- 
pose a  cord,  PAWBP',  to  pass 
over  them,  at  the  extremities  of 
which  are  suspended  the  two 
equal  weights  P,  P' ;  these  two 
weights  being  kept  in  equilibrio 
by  a  third  weight  W.*  Draw 
AE  parallel  to  CB,  and  produce 
WC  to  meet  it  in  E  ;  then  the  three  sides  C  A,  AE,  EC,  of  the  tri- 

*  In  this  and  the  following  articles  we  suppose  the  cords  to  be  without  weight, 
and  perfectly  inextensible  ;  and  the  pulleys  to  be  so  small,  and  so  adjusted,  that  the 
center  of  gravity  of  P,  P',  W,  &c.  may  all  lie  in  the  same  vertical  plane. 


MECHANICS.  125 

angle  ACE,  will  represent  the  quantity  and  direction  of  the  three 
forces  by  which  the  weight  W  is  kept  at  rest ;  for  these  three 
forces  are,  1st,  the  tension  produced  by  P  in  direction  CA;  2dly, 
the  tension  produced  by  P  in  direction  CB  or  AE  ;  and  3dly, 
its  own  gravity  in  the  direction  EC  ;  and  since  P'=P,  AE  or  CB 
must  be  equal  to  AC.  Hence 

P  or  P  :  W ::  AC  or  CB  :  EC  : :  sin.  AEC  :  sin.  ACB. 

153.  Join  AB  ;  then,  since  AB  is  parallel  and  CE  perpendicular 
to  the  horizon,  the  angles  at  D  are  right  angles,  and  since  ACE  and 
ACB  are  isosceles  triangles,  the  lines  AB,  CE,  bisect  each  other 
in  D.     Hence  EC=2CD.  and  P  :  W  : :  AC  :  (EC  or)  2CD ; 
.-.  2P  :  W  : :  2AC  :  2CD  : :  AC  :  CD,  and  4P2  :  W2  ::  AC2 : CD2 ;  also 

W2  v  A  TV 

4P2  — W2  :  W2  ::  AC2— CD2   (AD2)  :  CD8    /.CD8=^£^, 

or  CD=~F== ^;  which  gives  the  position  of  W  when   the 

equilibrium  takes  place,  for  P,  W,  and  AD  (iAB)  are  known 
quantities. 

Cor.  1.  If  P=£W,  or  2P=W,  then  4P2=W2,  and  4P2-W 
=0  ;  if  W  be  greater  than  2P,  then  4P2— W2  is  negative  ;  in  the 
former  case,  therefore,  the  value  of  CD  becomes  infinite,  and  in 
the  latter  impossible ;  which  shows  that  if  W  be  equal  to  or 
greater  than  2P,  no  equilibrium  can  take  place. 

Cor.  2.  If  W  and  AD  be  given,  CD  will  vary  inversely  as 
v/4P2  — W2;  and  if  P  be  indefinitely  increased,  W2  may  be 
neglected,  and  CD  will  vary  inversely,  as  \/  4P2=2P  ;  that  is, 
CD  will  diminish  as  the  sum  of  the  forces  is  increased,  but  can 
never  become  nothing  until  the  sum  of  the  forces  becomes  infi- 
nite. Suppose,  for  example,  the  line  ACB  to  be  a  rope  whose 
weight  is  equal  to  W  ;  then  it  would  require  an  infinite  force  to 
draw  it  into  a  straight  line,  by  powers  applied  at  its  extremities. 


CHAPTER  IX. 

OF  THE  MOTION  OF  BODIES  UPON  INCLINED  PLANES ;  AND  THE 
DOCTRINE  OF  THE  PENDULUM. 

154.  IN  the  second  chapter  we  investigated  the  relation  which 
takes  place  between  the  space  described,  the  velocity  acquired, 
and  the  time  of  its  motion,  when  a  body  ascends  or  descends 
perpendicularly  near  the  earth's  surface.  The  object  of  this 
chapter  is,  to  ascertain  that  relation  when  the  bodies  ascend  or 
descend  upon  planes  inclined  to  the  horizon. 


126  NATURAL   PHILOSOPHY 


MOTION    ON    INCLINED    PLANES. 

154.  In  Art.  130,  it  was  shown,  that  if  a 
weight  W  be  sustained  upon  an  inclined  f^  FiS- 104 
plane  AC,  (Fig.  104,)  by  another  weight  W 
acting  upon  it  in  a  different  direction  parallel 
to  the  plane,  then  W  :  W  : :  AB  :  AC.  Sup-w/ 
pose  now  the  string  WAW'  to  be  cut  in  two, 
then  it  is  evident  that  the  weight  W  would 
descend  down  the  plane  with  a  force  which 
bears  to  its  own  weight  the  ratio  of  AB  :  AC  ;  B 
and  since  (by  Art.  134)  this  force  is  constant  through  every  part 
of  the  plane,  the  body  thus  descending  may  be  considered  as 
acted  upon  in  every  point  of  its  descent  by  a  constant  force, 
which  bears  to  the  force  of  gravity  the  given  ratio  of  the  height 
of  the  plane  H  to  the  length  of  the  plane  L ;  call  this  force 
F,  and  let  the  force  of  gravity  be  represented  by  unity,  then 

F  :  1 : :  H  :  L,  and  F=^-;*  i.  e.  the  force  which  accelerates  the 

Li 
motion  of  a  body  down  an  inclined  plane  is  such  a  part  of  the 

force  of  gravity  as  may  be  represented  by  the  fraction  j- ;  this 

force,  therefore,  differs  not  from  the  force  of  gravity  in  kind,  but 
in  degree  ;  the  effects  produced  by  it  must  consequently  be  analo- 
gous to  the  effects  produced  by  gravity. 

156.  In  order  to  estimate  these  effects,  we  have  only  to  con- 
sider, that  if  a  body  be  acted  upon  by  different  constant  forces  for 
the  same  time,  the  velocity  generated  will  evidently  be  propor- 
tional to  the  intensity  of  those  forces  ;  and  that  if  it  be  acted 
upon  by  the  same  force  for  different  times,  the  velocity  will  be 
proportional  to  the  time  for  which  the  forces  act ;  from  which  it 
follows,  that  if  a  body  be  acted  upon  by  different  constant  forces 
for  different  times,  the  whole  velocity  generated  will  be  as  the 
force  and  time  conjointly.  |  Suppose  now  that  the  force  of 
gravity  is  represented  by  unity,  that  w=16TV  feet,  and  that  V= 
the  velocity  acquired  in  the  time  T  while  a  body  describes  the 
space  S  acted  upon  by  some  other  constant  force  F,  then,  from 
what  has  just  been  shown,  V  :  the  velocity  acquired  by  gravity 

in  l"=2m  (Art.  34)  : :  FxT  :  1  xl,  .-.  V=2wFT,  and  T=^p- 

Again,  since  V  x  F  xT,  T  x  V  must  vary  as  F  xT2  ;J  but  Sx  T  x  V 
(Art.  29,)  /.  S  x  F  xT2  ;  hence  S  :  the  space  described  by  gravity 

sin.  ACB 
*  Since  F :  1 : :  AB  :  AC  : :  sin.  ACB  :  rad.,  .-.  F=  — — j — oc  sin.  ACB  ;  therefore, 

the  force  which  accelerates  a  body  down  an  inclined  plane  varies  as  the  sine  of  the 
angle  of  the  plane's  elevation. 

t  Algebra,  420.  t  Algebra,  413. 


MECHANICS.  127 


(o  \  2 
-^j  .     Lastly, 

since  Vac  FxT,  V2  must  vary  as  FxTxV,  or  as  FxS,*  .-.  V2  : 
square  of  the  velocity  acquired  by  gravity  in  1"  (4m2)  :  :  F  xS  :  1  x 
space  described  by  gravity  inl"::FxS:lxwi;  hence  V2  =4mFS, 
V2 


,  and  S= 

157.  Let  us  now  apply  these  expressions  to  the  case  before  us, 
i.  e.  let  a  body  descend  down  an  inclined  plane  whose  height  is 

(H)  and  length  (L  ;)  then  F=^  (Art.  155  ;)  and  if  the  body  de- 
scends from  rest, 

TT  ,T    y 

Space  (S)  described  in  time  T=^xwT2,  and  T=(— 

Li  \  nin. 


TT 

Velocity  (V)  acquired  in  time  T=y-x2?wT,  and  T=  —  g  . 


/H         \ 

Vy  acquired  through  space  (S)  =2^X7^8  I  ,  and  S= 


LxV2 


—  . 

TT 

158.  Since   S=Tx?wT2,  and  H,  L,  m  are  given,  .'.  S  x  T2  ; 
Lt 

that  is,  the  space  described  varies  as  the  square  of  the  time  when 
a  body  falls  from  rest  down  an  inclined  plane,  as  well  as  when 
it  descends  freely  by  the  force  of  gravity  ;  the  spaces  described 
from  rest  in  equal  successive  portions  of  time  will  therefore  be 
as  the  odd  numbers  1,  3,  5,  7,  &c.  ;  and  if  the  body  be  projected 
upward  with  the  velocity  acquired  in  falling  through  any  space 
upon  the  plane,  it  will  ascend  to  the  point  from  which  it  fell,  the 
spaces  described  in  equal  successive  portions  of  time  being  as  the 
numbers,  1,  3,  5,  7,  &c.  taken  in  the  inverted  order.  If,  more- 
over, at  any  point  of  its  descent,  it  moves  forward  with  the  ve- 
locity acquired  continued  uniformly,  it  will  describe  twice  the 
space  in  the  same  time  as  that  in  which  it  has  fallen  to  acquire 
the  velocity  ;  and  if  it  be  projected  downward  or  upward  with 
the  velocity  (V,)  and  moves  for  the  time  (T,)  the  space  described 

in  that  time  will  be  equal  to  TxV±T  xmT2.     All  this  follows 

Lt 

from  the  law  of  acceleration  and  retardation  of  bodies  moving 
upon  inclined  planes,  being  the  same  as  that  which  regulates  the 
motion  of  bodies  descending  or  ascending  freely  by  the  force  of 
gravity.  (See  Arts.  31,  32,  and  33.) 

*  Algebra,  418,  cor. 


1S8  NATURAL   PHILOSOPHY. 

159.  The  expressions  contained  in  Art.  157,  apply  to  the  case 
of  a  body  descending  from  rest  through  any  part  of  an  inclined 
plane  whose  height  is  (H)  and  length  (L.)  If  the  body  falls 
through  the  whole  length,  then  S=FL,  .*.  the  velocity  acquired  in 

/H          \^ 
falling  down  the  whole  length  of  the  plane  =^(j  xmS\  = 

2\/mH=(by  Art.  34)  the  velocity  acquired  by  descending  freely 
through  the  height  H  ;  T-/^)  =-JL=  ;  but  the  time  of  fall- 


T 


/H\^ 
ing  freely  down  H  =1     j  ,  .-.  the  time  of  describing  the  whole 

length  of  the  plane  :  the  time  of  falling  freely  down  its  height 

/TT\ 

:f  —  I  :  :  L  :  H  :  :  length  of  the   plane  :  height   of  the 
R  \m> 
plane. 

Since  T=  ,  V=2\/wzH,  and  m  is  a  given  quantity,  T  va- 

VmH. 

Ties  as  -77^,  and  V  as  \/H;  i.  e.  the  time  of  describing  any  in- 
vH 

clined  plane  varies  as  its  length  directly,  and  as  the  square  root  of 
its  height  inversely;  and  the  velocity  acquired  varies  as  the 
square  root  of  the  height,  whatever  is  the  length  of  the  plane. 
Hence  we  deduce  the  following  THEOREMS. 

I.  The  VELOCITY  acquired  in  falling  down  an  inclined  plane,  i 
the  same  as  that  acquired  by  falling  freely  through  the  perpendicula- 
height  of  the  plane. 

II.  The  TIME  of  describing  the  whole  length  of  an  inclined  plane 
is  to  the  time  of  falling  freely  through  its  height,  as  the  length  of 
the  plane  to  its  height. 

III.  The  time  of  describing  any  inclined  plane  VARIES  AS   its 
length  directly,  and  as  the  square  root  of  its  height  inversely. 

IV.  The  velocity  acquired  in  describing  any  inclined  plane  VARIES 
AS  the  square  root  of  its  height,  whatever  be  the  length  of  the  plane.* 

160.  EXAMPLES. 

1.  How  far  will  a  body  descend  from  rest  in  4",  upon  an  in- 
clined plane  whose  length  is  400  feet,  and  height  300  feet  ? 

Ans.  193  feet. 

*  The  expressions  deduced  in  this  section  are  true  only  when  the  body  slides  down 
a  perfectly  smooth  plane  ;  for  in  this  case  it  is  evident  that  every  particle  of  the  body 
is  equally  accelerated,  and  therefore  whatever  is  proved  of  any  one  point  of  it  will 
apply  equally  to  all  ;  but  if  the  body  in  its  fall  has  a  rotary  motion  communicated  to 
it,  then  it  is  evident  that  all  the  points  of  it  will  not  be  equally  accelerated. 


m 

MECHANICS.  129 

2.  How  long  would  a  body  be  in  falling  down  100  feet  of  a 
plane,  whose  length  is  150  feet  and  height  60  feet? 

Ans.  3.94  seconds. 

3.  The  length  of  an  inclined  plane  is  60  feet,  and  its  eleva- 
tion 30°  :  What  velocity  would  a  body  acquire  in  falling  from 
rest  down  it  for  2"  ?  Ans.  32}  feet  in  1''. 

4.  The  height  of  a  plane  :  length  of  a  plane  :  :  7  :  15  :    How 
long  would  a  body  be  in  falling  down  it,  to  acquire  a  velocity  of 
20  feet  per  second  ?  Ans.  1.33  seconds.' 

5.  H  :  L  :  :  5  :  14  :  What  space  must  a  body  fall  through,  to 
acquire  a  velocity  of  10  feet  per  second  ?  Ans.  4.35  feet. 

6.  H  :  L  :  :  25  :  90  :  What  velocity  would  a  body  acquire  in 
falling  down  70  feet  ?  Ans.  35.37  feet  in  1". 

7.  The  length  of  an  inclined  plane  is  100  feet,  and  its  eleva- 
tion 60°  :  How  long  would  a  body  be  in  falling  down  it,  and 
what  velocity  would  it  acquire  at  the  end  of  its  fall  ? 

Ans.  T=2.68  seconds  ;  V=74.64  feet  in  1". 

8.  A  body  is  projected  up  an  inclined  plane  whose  length  is 
10  times  its  height,  with  a  velocity  of  30  feet  in  1"  :  In  what 
time  will  its  velocity  be  destroyed  ? 

Ans.  The  time  in  which  a  body  would  fall  down  an  inclined 
plane  of  this  elevation  to  acquire  a  velocity  of  30  feet  per  sec- 
ond=9.32". 

9.  A  body  is  projected  up  an  inclined  plane,  whose  height  is 
-}  of  its  length,  with  a  velocity  of  50  feet  per  second  :  Find  its 
place,  and  velocity  after  6"  are  elapsed  ? 

Ans.  S=203£  from  the  bottom  of  the  plane  ;  V=17|  feet  in  1". 

10.  A  body  falls  from  rest  down  the  inclined  plane  AC  (Fig. 
105  :)  Compare  the  times  of  describing  the  first  and  last  halves  of  it. 

Bisect  AC  in  D,and  draw  DE  parallel  to  CB  ;  by  Art.  159,  the 

time  down  AC  :  the  time  down  AD  :  :  ^—  —  :    .—  -  :  :  (by  similar 

AC        AD 

triangles)     =  :     =  :  :  ^AC  :  ^AD  :  :  V2  :  1.     Hence, 


Fig.  106. 
Fig.  105.  A 


C 
17 


130  NATURAL   PHILOSOPHY. 

(1.)  The  times  down  different  parts  of  the  same  inclined  plane 
vwhen  the  body  falls  from  rest  from  the  top  of  the  plane)  are  to 
each  other  as  the  square  roots  of  the  lengths  of  those  parts. 

(2.)  The  time  down  AC  —  the  time  down  AD  (i.  e.  the  time 
down  DC)  :  the  time  down  AD  :  :  \/2—  1  :  1. 

11.  To  mark  out  upon  a  plane  AC  (Fig.  106,)  a  part  ED 
which  shall  be  equal  to  the  height  AB,  and  which  a  body  (falling 
down  AC)  would  describe  in  the  same  time  as  one  falling  freely 
through  AB. 

Let  AC=a      ^ 
ABorED=6      1      T  AD*  :  TAE  :  :  VM)  :  VAE 


then  AD=b+x  J 
.'.  TAD—  TAE  (i.  e.  TDE)  :  TAE  :  : 

but  TAE  :  TAG  :  :  VAE~:  v"AC":  :  Jx  :  </a, 

and  by  Art.  159,  TAG  :  TAB  :  :  a  :  b  :  :  Va  :  —  -  ; 


f  TDE  :  TAB  :  :  Vb+x-V*  :  -f 
v« 

But  TDE=TAB,  .: 


which  equation  solved  gives  x=  -  -  or  AE=  --  .  _     . 

4a  4AC 

161.  QUESTIONS  ON  THE  INCLINED  PLANE. 

1.  The  length  of  an  inclined  plane  is  480  feet,  and  the  height 
210  ;  a  body  falls  from  rest  from  the  top  of  the  plane  :  What 
space  will  it  have  fallen  through  in  6"  ;  what  time  will  it  be  in 
falling  through  450i  feet  ;  and  what  velocity  will  it  have  ac- 
quired, when  it  has  arrived  within  124.7  feet  of  the  bottom  of 
the  plane  ? 

Ans.  S=253fV/ee*;  T=8  seconds;  V  =100  feet  in  1". 

2.  A  body  has  been  falling  for  15"  down  an  inclined  plane 
whose  length  is  2£  times  its  height  :  What  velocity  will  it  have 
acquired  at  the  end  of  its  fall  ?  Ans.  V=193  feet  in  1". 

3.  The  elevation  of  a  plane  is  30°  ;  a  body  in  falling  from  the 
top  to  the  bottom  of  it,  acquires  a  velocity  of  579  feet  in  1"  : 
What  is  the  length  of  the  plane  ?  Ans.  L=10422/ee£. 

4.  A  car  broke  loose  from  the  top  of  an  iron  railway  eleven 
miles  in  length,  which  was  uniformly  inclined  to  the  horizon  at 
an  angle  of  1  degree.     Supposing  the  car  to  move  without  re- 

*  TAD  means  the  time  down  AD,  and  so  of  the  rest.  t  Alg.  393. 


MECHANICS.  131 

sistance,  in  what  time  would  it  reach  the  lower  end  of  the  rail- 
way, and  what  velocity  would  it  acquire  ? 

Am.  T=7'34".88;  V=255.36/ee«per  second. 

5.  At  Alpnach,  in  Switzerland,  is  a  celebrated  slide  for  con- 
veying timber  tre*es  from  Mount  Pilatus  to  Lake  Luzerne,  whence 
they  are  transported  down  the  Rhine.     The  slide  consists  of  an 
inclined  plane  formed  of  logs  in  the  shape  of  a  trough,  into 
which  the  trees  are  launched,  and  down  which  they  descend  by 
the  force  of  gravity.     It  is  8  miles  in  length,  and  is  inclined  to 
the  horizon  on  an  average,  at  an  angle  of  3°  14' :  In  what  time 
will  a  tree  descend  from  the  top  to  the  bottom  of  this  plane,  no 
allowance  being  made  for  friction  ?  Ans.  3'  35".8 

6.  Trees  descending  the  slide  sometimes  "  bolt  out"  of  the 
trough,  and  occasion  great  destruction :  With  what  force  would 
a  tree  weighing  1500  pounds,  leaping  out  of  the  slide  at  the  end 
of  7  miles,  strike  upon  an  obstacle,  as  for  example,  a  standing 
tree  ?  Ans.   With  a  force  equal  to  549318.2  Ibs. 

7.  Two  inclined  planes  have  a  common  Height  of  75  feet ;  the 
elevation  of  one  of  them  is  50°,  and  of  the  other  20°  :  With  what 
velocity  must  a  body  be  projected  from  the  bottom  of  the  former, 
that  it  may  just  rise  to  the  top  of  the  latter ;  and  what  will  be 
the  whole  time  of  its  ascending  and  descending  through  the  two 
planes  ?*  Ans.  ~V=69AQfeet  in  1" ;  T=9.133  seconds. 

8.  How  long  will  a  body  be  in  falling  down  the  last  half  of  a 
plane,  whose  height  is  1  mile,  and  angle  of  elevation  1  minute  ? 

Ans.  T=5  Jiours,  4  minutes,  3  seconds. 

9.  The  length  of  a  plane  is  250  feet,  and  height  150  :  Mark 
out  upon  it  a  part  equal  to  the  height  which  a  body  in  falling 
down  it  describes,  while  another  body  would  descend  freely 
through  the  height. 

Ans.  It  begins  to  describe  this  part  when  it  has  fallen  through  10 
feet  from  the  top  of  the  plane. 


MOTION  OF  BODIES  DOWN  DIFFERENT  SYSTEMS  OF  INCLINED  PLANES. 

162.  It  has  already  been  shown,  that  when  a  body  descends 
down  an  inclined  plane  whose  length  is  L,  and  height  H,  the  ve- 
locity acquired  varies  as  \/H,  and  the  time  of  description  as  — -  ; 

•s/H 

let  us  now  apply  these  expressions  to  finding  the  relation  between 
the  times  and  velocities  of  bodies  falling  down  different  systems 
of  inclined  planes. 

163.  Let  AC,  AD,  AE,  (Fig.  107,)  &c.,  be  a  system  of  inclined 
planes  having  the  same  height,  AB  ;  then  since  the  velocities  ac- 

*  The  planes  are  placed  as  in  Fig.  92 ;  a  body  is  projected  from  D  with  a  velocity 
just  sufficient  to  carry  it  to  A,  and  then  falls  from  rest  down  the  plane  AC. 


132 


NATURAL   PHILOSOPHY. 


quired  by  bodies  falling  down  these  planes  are  as  VAB,  and  the 

AC     AD     AE 
times  of  description  as     —  ,      —.  =  ,     ~rr   &c.  (i.  e.  as  AC,  AD 


AE,  &c.)  it  is  evident  that  bodies  falling  down  a  system  of  planes 
of  this  kind  would  acquire  at.the  end  of  their  fall  the  same  ve- 
locity,* and  that  the  times  of  description  would  be  as  their  re- 
spective lengths. 

A  Fig.  108 

Fig.  107.  ^  A 


164.  If  chords  be  drawn  in  a  circle  from  the  extremity  of  that 
diameter  which  is  perpendicular  to  the  horizon,  the  VELOCITIES  which 
bodies  acquire  in  falling  through  them,  are  proportional  to  their 
lengths  ;  and  the  TIMES  of  describing  these  chords  are  all  equal  to 
one  another,  and  are  severally  equal  to  the  time  of  describing  the 
diameter. 

Let  the  diameter  AB  of  the  circle  ACB  (Fig.  108,)  be  perpen- 
dicular to  the  horizon  ;  draw  the  chords  AC,  AD,  AE,  and  CB, 
DB,  EB  ;  draw  also  Cc,  Dd,  Ee,  &c.,  parallel  to  the  horizon ;  then 
the  velocities  acquired  by  bodies  falling  down  the  former  system 
of  chords  are  as  \/Ac,  VAd,  VAe,  and  down  the  latter  as  VcB, 

— •       —  ,  AC2  ADS  CB2 

but     Ac=  -TrJ't    Ad—  AP  ;    and    cB=-rr»'    dB= 
AD  AD  AD 


DB2 


;  hence  VAc,  VAd,  </Ae,  vary  as  AC,  AD,  AE,  and 


J,  VeB,  vary  as  CB,  DB,  EB.     The  times  of  describing  AC, 

AD,  AE,  and  CB,  DB,  EB,  are  as  -7=,  -7==,  -=,  and  -= 

v'Ac  ^Ad  ^Ae  vcB 

DB     EB  t 

•  ,—— , '  ,— •;  but  each  of  these  quantities  is  equal  to  \/AB.I 

*  And  by  Art.  159,  this  velocity  is  equal  to  the  velocity  which  a  body  would  ac- 
quire by  falling  freely  through  the  height  (AB)  of  the  plane. 

t  For  (Euc.  8.  6.)  Ac  :  AC  : :  AC  :  AB,  or  Ac=A^. 
AP*  A  r> 

t  For  since  Ac  :  AC  : :  AC  ,:  AB/-— -=AB, .:~  =  v'AB ;  and  so  of  the  rest. 
Ac  vAc 


MECHANICS.  133 

165.  Suppose    now  that  a    body       Fig.  109.       A    E 
falls  down  a  system  of  planes  AB, 

BC,  CD,  (Fig.  109,)  inclined  to  each 

other,  and  that  no  velocity  is  lost  in 

falling  from  one  plane  to  the  other. 

Draw  AF   parallel    to  the  horizon, 

and  produce  DC,  CB  to  meet  it  in  F, 

E ;  through  B,  C,  draw  Bb,  Cc,  par-  ^ 

allel  to  AF,  and  jet  fall  FG  at  right D 

angles  to  the  horizon.     By  Art.  163,  the  velocity  down  AB=ve- 

locity  down  EB=velocity  down  Fb ;  .'.the  velocity  down  AB+ 

BC=velocity  downEB+BC  (or  EC)=velocity  down  FC=velocity 

down  Fc ;  and  reasoning  in  the  same  manner,  the  velocity  down 

AB+BC+CD=velocity  down  FD=velocity  down  FG;  i.  e.  the 

whole  velocity  acquired  by  a  body  falling  down  successive  planes, 

is  equal  to  the  velocity  which  a  body  would  acquire  in  falling 

freely  through  their  joint  height  FG. 

166.  If  the  number  of  planes  be  increased  &S- 110< 
infinitely,  the  figure  will  become  a  curve,  as 

in  Fig.  110;  and  hence  the  velocity  ac- 
quired in  descending  through  any  perfectly 
smooth  curved  surface,  is  the  same  as  that 
acquired  by  falling  through  the  perpendicu- 
lar height  of  the  curve.  __ 

Hence,  in  general,  a  body,  by  descending  D G 
from  a  certain  height  to  the  same  horizontal  line,  will  acquire  the 
same  velocity,  whether  the  descent  be  made  perpendicularly  or  ob- 
liquely, over  an  inclined  plane,  or  over  many  successive  inclined 
planes,  or  over  a  curve  surface.* 

167.  The  times  and  velocities  of  bodies  falling  down  planes  simi- 
larly inclined  to  the  horizon,  are  to  each  other  both  as  the  square  roots 
of  the  lengths,  and  as  the  square  roots  of  the  heights  of  the  planes. 


C  B    c  b 

Let  AC,  ac,  (Fig.  Ill,)  be  two  planes  similarly  inclined  to  the 
horizon,  then  AC  :  ac  :  :  AB  :  ab,  and  VAC  :  \/ac  :  :  -/AB  :  <Jab. 

AC        ac 
Now  the  time  down  AC  :  the  time  down  ac  :  : 


*  Cavallo. 


134 


NATURAL   PHILOSOPHY. 


AC        ac  _  __        _ 

•/-—  :  -7=  : :  VAC  :  Vap,  or  V  AB  :  Jab ;    and  the   velocity 
"  AU     "  flc 

acquired  in  falling  down  AC  :  the  velocity  down  ac  : :  JAB  : 
Jab,  or  \/AC  :  \/ac. 

168.  The  times  of  descent  down  SIMILAR  SYSTEMS  of  inclined 
planes  are  as  the  square  roots  of  the  lengths  of  the  planes.* 

Fie.  112. 

E*      F  • 


D  G    d  g 

Let  there  be  two  systems  of  planes,  AB,  BC,  CD,  ab,  be,  cd, 
(Fig.  112,)  similar,  and  similarly  situated  with  respect  to  the 
horizon,  and  complete  the  figures  as  in  Art.  164  ;  then,  supposing 
no  velocity  to  be  lost  in  passing  from  one  plane  to  the  other,  and 
using  the  notation  employed  on  page  130,  we  have,  by  Art.  167, 
TAB  :Tab::  v/AB~:  Jab.  (X.) 

TEC  :  Tec  :  :  VEC  :  Jec  :  :  JAB  :  JabJ 
TEB  :  Teb  :  :  JEB  :  Jeb  :  :  JAB  :  Jab; 
Hence, 

TEC  -TEB  (or  TBC)  :  Tec-Tel  (or  Tbc)  :  :  J~AB  :  Jab.    (Y.) 
In  the  same  manner  it  may  be  shown,  that 

TCD  :Tcd::  J~AB  :  -Jab.  (Z.) 

From  the  proportions  marked  (X,)  (Y,)  (Z,)  therefore,  we  have 
T  (AB+BC+  CD)  :  T  (ab+bc+cd)  :  :  JAB  :  Jab 

::  V  (AB+BC+CD)  :  J(ab+bc+cd).l 

169.  The  times  of  descending  through  SIMILAR  CURVES,  similarly 
situated  with  respect  to  the  horizon,  are  as  the  square  roots  of  the 
lengths  of  those  curves. 

If  the  number  of  planes  AB,  BC,  CD,  &c.,  ab,  be,  cd,  &c.,be  in- 


*  The  velocity  down  AB+BC+CD  :  velocity  down  ab+bc+cd  :  :  v/FG  :  </fg  :: 
JFD  :  </fd  :  :  ,/A3  :  Jab  :  :  (AB+BC+CD)  :  J  '(ab+bc+cd).  The  velocities 
as  well  as  the  times  are  therefore  as  the  square  roots  of  the  lengths  of  the  planes. 

t  For  by  similar  triangles,  EB  :  eb  :  :  AB  :  ab  ;  and  from  similar  planes  BC  : 
icjj_AB  :ab_  ;  .-.EB+BC  (or  EC)  :  eb+bc  (ec)  :  :  AB  :  ab  ;  and  JEC  :  ^7c~:  : 
V  AB  :  v/  ab.  In  the  same  manner  it  may  be  shown  that  FD  :fd::  AB  :  ab,  or 
•V/FlI:  l/fd:  :  »/AB  :  Jab. 

t  For  since  AB_:  «6  :  :  BC  :  be  :  :  CD  :  cd,  AB+BC+CD  :  ab+bc+cd  :  :  AB  : 
ab  ;  or  ^AB  :  Jab  :  :  J(  AB+BC+CD)  :  (ab+bc+cd). 


MECHANICS. 


135 


creased,  and  their  lengths  and  their  inclinations  to  each  other  be 
diminished  ad  infinitum,  then  the  polygons  ABCD,  abed,  become 
similar  curves,  in  falling  down  which  no  velocity  is  lost. 

Suppose  the  curves  to  be  circular  arcs ;  then,  since  similar 
circular  arcs  are  to  each  other  as  the  radii  of  the  circles  to  which 
they  belong,  the  times  of  descending  through  these  arcs  will  be 
to  each  other  as  the  square  roots  of  their  radii. 


THE    PENDULUM. 

170.  DEFINITIONS. — A  Pendulum  is  a  body  suspended  by  a  right 
line  from  any  point,  and  moving  freely  about  that  point  as  a 
center.  The  point  about  which  the  pendulum  revolves,  is  called 
the  center  of  suspension.  The  vibration  of  a  pendulum,  is  its  mo- 
tion from  a  state  of  rest  at  the  highest  point  on  one  side,  to  the 
highest  point  on  the  other  side.*  The  center  of  oscillation  of  a 
pendulum,  is  such  a  point,  that,  were  all  the  matter  of  the  pen- 
dulum collected  in  it,  the  quantity  of  motion  (or  momentum) 
would  be  equal  to  the  sum  of  the  momenta  of  all  the  parts  taken 
separately. 

Thus,  (Fig.  113,)  the  parts  of  the  pendu- 
lum about  b  move  faster  than  those  about  a, 
and  consequently  have  more  momentum  ;  but 
there  is  a  point  about  which  the  momenta 
balance  each  other,  and  therefore  in  the  in- 
vestigations relating  to  the  pendulum,  all  the 
parts  of  which  it  consists  may  be  considered 
as  concentrated  in  that  point. 

The  center  of  oscillation  is  below  the  cen- 
ter of  gravity ;  for,  since  the  parts  more  re- 
mote from  the  center  of  suspension  have 
more  velocity  than  the  parts  that  are  nearer 
to  it,  the  quantity  of  matter  below  the  center 
of  oscillation  must  be  less  than  the  quantity 
of  matter  above  it. 


171.  In  order  to  understand  the  doctrine  of  the  pendulum,  it 
is  necessary  to  become  acquainted  with  a  few  of  the  leading 
properties  of  the  curve  called  the  Cycloid.f 

*  In  these  investigations,  as  in  those  of  the  Mechanical  Powers,  pendulums  are 
supposed  to  move  without  any  resistance  from  the  air  or  from  friction.  The  con- 
clusions, therefore,  will  be  accurately  true  only  when  applied  to  vibrations  per- 
formed in  a  perfect  vacuum,  round  a  perfectly  smooth  axis  of  suspension. 

t  The  learner  will  remark  that  the  mode  of  reasoning  is  this :  it  is  first  proved  that, 
were  a  pendulum  to  vibrate  in  a  cycloid,  all  its  vibrations,  whether  performed  in  larger 
or  in  smaller  arcs,  would  be  equal.  It  is  then  shown  that  pendulums  vibrating  in 
small  circular  arcs  are  subject,  very  nearly,  to  the  same  law.  The  calculus  affords 
an  easier  method  than  Geot«try  of  investigating  the  properties  of  this  as  well  as  of 


136 


NATURAL   PHILOSOPHY. 


A  Cycloid  is  the  curve  described  by  a  point  in  the  circumference 
of  a  circle  rolling  in  a  straight  line  on  a  plane. 


Fig.  114. 


Let  the  circle  AHB  (Fig.  114)  make  one  revolution  upon  the 
line  CAX,  equal  to  its  circumference  ;  the  curve  line  CDBX, 
traced  out  by  that  point  of  the  circle  which  was  in  contact  with 
C  when  the  circle  began  to  revolve,  is  called  a  Cycloid.  If  CX 
be  bisected  in  A,  and  AB  be  drawn  at  right  angles  to  it,  it  is  evi- 
dent from  the  manner  in  which  the  curve  is  generated,  that  it 
will  have  similar  branches  on  both  sides  of  AB,  and  that  its 
vertex  B  will  be  so  placed  as  to  make  its  axis  AB  equal  to  the  di- 
ameter of  the  generating  circle.  Its  properties,  as  applied  to  the 
vibration  of  the  pendulum,  are  the  following. 

172.  The  cycloidal  ordinate  DH  equals  the  circular  arc  BH. — 
For,  let  bDa  (Fig.  114)  be  the  position  of  the  circle  when  the 
generating  point  is  at  D  ;  draw  the  diameter  ba  parallel  to  BA, 
and  from  D  draw  DHL  parallel  to  CA  ;  then  the  arc  Da=arc  HA, 
.'.  sin.  DO=sin.  HL,  and  consequently  DH=OL;  but  from  the 
mode  in  which  the  cycloid  is  generated,  Ca=arc  Da,  and  CA— 
semicircle  BHA ;  hence  DH=OL=aA=CA  —  Ca=semicircle 
BHA-arcHA=arcBH. 


Fig. 


173.  A  tangent  to  the  cycloid  at  any 
point,  E,  (Fig.  115,)  is  parallel  to  the 
corresponding  chord  BK  of  the  gener- 
ating circle. — Draw  DHL  indefinitely 
near  to  EKM  ;  join  BK,  and  produce 
it  to  k  ;  let  fall  Ho  at  right  angles  to 
K&.  The  indefinitely  small  triangle 
HKA;  is  similar  to  the  triangle  KRB 
formed  by  the  tangents  (KR,  BR)  to  C 
the  circle  at  the  points  K,  B,  and  is  consequently  isosceles  ; 
.•.KH=H&.  Now  by  Art.  172,  arc  BKH=DH, .-.  BKH-KH 
(=arc  BK=EK)  =DH-HA=DA- ;  but  since  EK  and  Vk  are 
equal  and  parallel,  ED  and  Kk  must  also  be  equal  and  parallel ; 
and  as  the  tangent  at  the  point  E  may  be  considered  as  coin- 
ciding with  ED,  it  must  therefore  be  parallel  to  the  chord  BK. 


other  curves ;  but  the  geometrical  method  is  retained  in  this  treatise,  to  render  the 
study  intelligible  to  such  as  are  not  acquainted  with  the  Calculus. 


MECHANICS. 


137 


174.  The  cycloidal  arc  BE  is  equal  to  twice  the  corresponding 
chord  BK  of  the  generating  circle. — For  since  the  triangle  KH.k 
is  isosceles,  Ho  bisects  the  base  Kk,  .-.  Kk  or  ED=2Ko ;  and 
since  Ho  may  be  considered  as  a  small  circular  arc  described  with 
radius  BH,  Ko=Bo-BK=BH— BK;  hence  ED  and  Ko  are  co- 
temporaneous  increments  of  the  cycloidal  arc  BE  and  the  chord 
BK ;  and  as  the  arc  and  chord  begin  together  from  the  point  B, 
and  the  former  increases  by  ED  or  Kk  while  the  latter  increases 
by  Ko=iED,  the  arc  BE  must  be  equal  to  twice  the  chord  BK ; 
consequently,  the  whole  arc  BC=twice  the  diameter  AB. 

175.  AHB  (Fig.  116)  is  a  circle,       _  Fig.  116. 
whose  diameter  AB  is  perpendicular 

to  the  horizon,  and  CDB  a  cycloidal 
arc.  Now,  by  Art.  159,  if  a  body 
begins  to  descend  from  any  point 
D,  its  velocity  at  the  point  E  will 
be  the  same  as  the  velocity  at  the 
point  M  of  a  body  falling  freely 
through  the  perpendicular  height 
LM ;  and  its  velocity  at  every  other 
point  during  its  descent  through  the 
cycloidal  arc  DEB  will  be  the  same 
as  the  velocity  of  the  body  falling 
freely  at  every  other  corresponding 
point  of  the  line  LMB.  By  Art. 
167,  therefore,  the  velocity  V  of  the 
body  thus  descending  along  the  cy- 
cloidal arc  DE  will  vary  as  \/(LM) 
QC  V  (EL  -  BM)x  V  (ABxBL  -  ABxBM)x  V  (HB2-KB2)  oc  V 
(DB2-EB2.)  Hence,  let  Ed  be  drawn  parallel  to  AC  and  equal 
to  BD,  and  upon  Ed  describe  the  quadrant  of  a  circle  dmN  ;  take 
Be  equal  to  BE,  and  draw  em  at  right  angles  to  Ed ;  then  will 
em2=Bm2-Be2=Bt/2-Be2=DB2-EB2,  and  consequently  V,  which 
varies  as  #(DB2— EB2,)  will  vary  as  em,  the  sine  of  the  arc  dm, 
whose  versed  sine  is  de  or  DE,  the  space  fallen  through. 

176.  Let  EF  be  an  indefinitely  small  part  of  the  cycloidal  arc, 
and  make  ef  equal  to  it ;  draw/«  at  right  angles  to  Ed,  and  mo 
parallel  to  it.     Since  EF  .is  very  small,  it  may  be  considered  as 
described  with  the  velocity  V  at  E  continued  uniformly,  and 

therefore  the  time  of  describing  EF  [since  T=^jwill  be  repre- 

"C*T71  /» 

sented  by  -^-or— .  Now  since  the  sine  em  represents  the  velo- 
city at  any  point  E,  the  whole  velocity  acquired  in  falling  down 
DB  will  be  represented  by  the  radius  BN  or  Em  ;  if,  therefore, 
a  body  were  to  describe  the  quadrantal  arc  dmN  with  the  velo- 

18 


138 


NATURAL    PHILOSOPHY. 


city  at  the  lowest  point  continued  uniformly,  the  time  of  describ- 
ing any  small  part  (mri)  of  it  would  be  represented  by  m-^.  But 

tjtn 

by    similar    triangles,   Bme,   mno,   Em  :  em  :  :  mn  :  mo  or    ef, 

.'.  —  =  ^— ;  hence  the  time  of  describing  the  small  cycloidal  arc 
em     Bm 

EF  is  equal  to  the  time  of  a  body's  moving  through  the  corres- 
ponding small  circular  arc  mn,  with  the  velocity  in  B  continued 
uniformly ;  the  whole  time  of  descent  therefore  through  DEB,  will 
be  equal  to  the  time  of  a  body's  describing  the  quadrantal  arc  dmN 
with  the  velocity  at  B  continued  uniformly. 

177.  Now  the  velocity  at  the  lowest  point  B  of  the  cycloid  is 
equal  to  the  velocity  acquired  in  falling  down  the  chord  HB,  and 
by  Art.  31,  with  this  velocity  continued  uniformly  it  would  de- 
scribe 2HB=BD=Brf  in  the  same  time.  But  by  Art.  164,  the 
time  of  falling  down  the  chord  HB=the  time  of  falling  down 
the  diameter  AB  or  the  axis  of  the  cycloid ;  hence  the  time  of 
descending  down  the  cycloidal  arc  DB,  and  the  time  of  falling 
freely  through  the  axis  AB  of  the  cycloid,  are  to  each  other  as 
the  times  of  describing  the  arc  dmN  and  the  straight  line  Bd 
with  the  same  uniform  velocity,  i.  e.  as  the  quadrant  of  the  circle 
to  its  radius.* 


Fig.  11 7. 
E 


S 


178.  Take  any  line  SC,  (Fig.  117,) 
and  draw  SA  at  right  angles  to  it ; 
make  SC  :  SA  : :  semi-circumference  of 
a  circle  :  its  diameter  ;  and  complete 
the  parallelogram  SCDA.  Produce 
SA  to  B,  making  AB=SA  ;  upon  SC, 
AD,  describe  two  semi-cycloids  SD, 
DB,  the  vertex  of  the  former  of  which 
is  at  D,  and  the  latter  at  B ;  then  if 
a  body  be  suspended  from  a  point  S 
by  a  string  whose  length  is  equal  to 
the  cycloid  SD,  and  begins  to  descend 
from  D,  its  place  will  always  be  in  the 
semi-cycloid  DB,  the  part  TP  of  the 
string  being  always  at  right  angles  to 
the  cycloidal  arc  DB.  For  through  any  point  F,  draw  EFG  per- 
pendicular to  SC,  and  through  B  draw  BG  parallel  to  SC ;  then 
EG=SB  ;  on  EF,  FG,  describe  the  two  semicircles  ETF,  FPG, 
and  draw  the  chords  TF,  FP,  the  former  of  which  (Art.  173)  is  a 
tangent  to  the  cycloid  SD  in  T.  Now  SE=arc  ET,  and  SC= 
ETF,  .-.  CE(=DF)=arc  TF=arc  FP ;  hence  the  angles  TEF, 
FGP  are  equal,  and  consequently  the  triangles  TEF,  FGP  similar 

*  In  uniform  motions,  when  the  velocities  are  equal,  the  times  are  as  the  spaces 
(Art.  12.) 


MECHANICS.  139 

and  equal  to  each  other ;  .-.  TF,  FP,  are  in  the  same  straight 
line ;  moreover,  TP=2TF=(Art.  174)  the  cycloidal  arc  TD  ;  if, 
therefore,  the  string  TP  be  always  equal  to  the  cycloidal  arc  TD, 
i.  e.  if  the  whole  string  STP  be  equal  to  the  semi-cycloid  SD,  P 
will  always  be  found  in  the  cycloid  DB  ;  and  since  the  angle 
TPG  is  a  right  angle,  and  PG  (Art.  173)  is  a  tangent  to  the  cy- 
cloid, therefore  TP  is  always  at  right  angles  to  the  curve. 

179.  A  pendulum  may  be  made  to  vibrate  in  a  cycloid,  by  causing 
the  flexible  rod  or  string  to  apply  itself,  as  it  moves,  to  the  sides  of 
two  semi-cycloids. 

S 
Fig.  118. 


Instead  of  supposing  the  body  to  descend  down  a  curve  in  the 
form  of  a  cycloid,  the  effect  will  evidently  be  the  same  with  re 
spect  to  the  acceleration  of  the  body,  whether  it  be  kept  in  a 
curve  of  this  form  by  the  reaction  of  the  surface  in  a  direction 
perpendicular  to  it,  or  by  the  tension  of  a  string  acting  in  the 
same  direction.  Now  a  small  body  P,  (Fig.  118,)  suspended 
from  the  point  S  by  a  string  STP  of  the  same  length  with  either 
of  the  semi-cycloids  SD,  SE,  and  made  to  vibrate  between  them, 
(the  string  gradually  unwinding  from  the  semi-cycloid  SD  as  it 
descends  to  the  lowest  point  B,  and  winding  round  the  semi-cy- 
cloid SE  as  it  ascends  to  the  highest  point  E,)  will  always  be 
found  in  a  cycloid  similar  and  equal  to  the  two  semi-cycloids  SD, 
SE  ;  and  as  the  string  TP  is  always  at  right  angles  to  the  curve, 
it  will  be  under  precisely  the  same  circumstances  as  the  body 
descending  down  the  curve.  Having  descended  to  the  lowest 
point  B,  it  will  then  ascend  through  an  arc  BE,  equal  to  BD,  in 
the  same  time  in  which  it  descended  down  DB  ;  so  that  the 
whole  time  of  one  vibration  will  be  twice  the  time  of  descend- 
ing down  DB  ;  hence,  (Art.  177,)  the  time  of  one  vibration  will  be 
to  the  time  of  a  body's  falling  freely  down  half  the  length  of  the  pen- 
dulum, as  the  circumference  of  a  circle  to  its  diameter.  -~" 

180.  Since  (Art.  179)  the  time  of  a  vibration  is  to  the  time 
down  the  axis  in  the  same  given  ratio,  whatever  be  the  point 


140 


NATURAL    PHILOSOPHY. 


from  which  the  pendulum  begins  its  oscillations,  all  the  vibrations 
of  a  pendulum  of  the  same  length  are  equal  to  each  other.  But 
from  the  difficulty  of  constructing  plates  of  the  exact  form  SD, 
SE,  and  from  other  causes,  the  cycloidal  pendulum  is  of  little  or 
no  practical  utility.  In  a  geometrical  point  of  view,  however, 
this  mode  of  comparing  the  time  of  a  vibration  with  the  time  of 
falling  down  a  space  equal  to  half  the  length  of  the  string,  is  of 
considerable  importance  ;  for  the  cycloid  at  the  lowest  point  B 
may  evidently  be  considered  as  a  circular  arc  described  with  the 
radius  SB  ;  if  therefore  a  body  be  suspended  by  a  string  whose 
length  is  SB,  and  vibrates  in  a  circular  arc  only  to  a  very  short 
distance  on  each  side  of  the  point  B,  the  time  of  the  vibration  of 
a  pendulum  in  small  CIRCULAR  arcs,  is  to  the  time  down  half  the 
length  of  the  pendulum,  as  the  circumference  of  a  circle  to  its  di- 
ameter ;  and  therefore,  within  moderate  limits,  the  time  will  be  the 
same,  whether  the  arc  of  vibration  be  larger  or  smaller. 

181.  The  times  of  vibration  of  pendulums  of  different  lengths  are 
to  each  other  as  the  square  roots  of  the  lengths. 

Let  L=length  of  the  string  or  thin  inflexible  rod  by  which  a 
small  body  is  suspended,  tf=3.14159,  &c.  m=lQ^  feet;  then, 
(Art.  34)  the  time  of  a  body's  falling  down  half  the  length  of 

(Q  v    5         x  j     v   2 
—  1   =  f  —  j  .     Hence  the  time  of  a  vibration  (T) 

*  :  !»  or  T^  Or"}  »  wnich  varies  as  ^L-     Let  T=l  : 


/*2L\       .  2m      32.1666 

then  (  ^  \  •  =1,  or  **L=2m,  and  L=  -^  =  =3.259   feet= 

39.11  inches;  if  the  space  fallen  through  from  rest  by  gravity  in 
1",  therefore,  be  16TL  feet,  the  length  of  a  pendulum  which  vi- 
brates seconds  will  be  39.11  inches. 

Let  ,r=the  space  fallen  through  by  gravity  in  1",  then  T= 

(TT-  )    and  T2  —-n~t  or  ****3jiJ  ^  therefore  the  length  of  a 

pendulum  which  vibrates  in  T"  is  given,  the  space  fallen  through 
by  gravity  in  1"  may  be  found  ;  thus,  let  the  length  of  the  pen- 
dulum which  vibrates  seconds=39.2  inches,  then 
L      9.8739.2 


182.  The  times  of  vibration  of  pendulums  of  different  lengths 
acted  upon  by  DIFFERENT  ACCELERATIVE  FORCES,  will  vary  as  the  square 
roots  of  the  lengths  directly,  and  as  the  square  roots  of  the  forces 
inversely. 


MECHANICS.  141 


If  the  accelerative  force  be  not  given,  then  (Art.  156)  the  tim» 

(S  \       /    L  \ 
~F/       \2~F/  * 


* 


=  1,  °r  T= 


which  varies  as(|) 


The  times  of  vibrations  of  pendulums  of  the  SAME  LENGTH  vary 
inversely  as  the  square  roots  of  the  accelerative  forces. 

If  L  be  given,  then  Tx  —  —  . 

The  lengths  of  pendulums  vibrating  in  the  SAME  TIME  vary  as 
the  forces  which  accelerate  them 
If  T  be  given,  then  L  oc  F. 

183.  The  NUMBER  of  vibrations  performed  in  a  given  time  by  pen- 
dulums of  different  lengths,  acted  upon  by  different  accelerative 
forces,  are  directly  as  the  square  roots  of  the  forces,  and  inversely 
as  the  square  roots  of  the  lengths. 

Let  n=number  of  vibrations  performed  in  any  given  time,  and 
T=time  of  one  vibration  ;  then  n  will  vary  inversely  as  T= 


If  therefore  the  lengths  be  given,  the  number  of  oscillations 
will  be  directly  as  the  square  roots  of  the  forces  ;  and  if  the  forces 
be  given,  the  number  of  oscillations  will  be  inversely  as  the  square 
roots  of  the  lengths. 

184.  EXAMPLES.  • 

1.  What  is  the  time  of  an  oscillation  of  a  pendulum  whose 
length  is  10  feet ;  and  what  must  be  the  length  of  a  pendulum 
which  shall  oscillate  ten  times  a  minute  ? 

The  length  of  a  pendulum  which  oscillates  in  1"  is  39.11 
inches  ;  and  (Art.  181)  the  times  of  vibration  of  pendulums  of 
different  lengths,  are  to  each  other  as  the  square  roots  of  their 
lengths ;  hence  1"  :  T : :  VB9.ll  :  ^120  : :  625  :  1095,  T=W/  = 
IfH  seconds. 

Again,  let  T=6"=time  of  one  oscillation  of  a  pendulum  which 
makes  ten  in  a  minute,  then  6"  :  1"  ::  ^L  :  V39.ll,  .'.  36  :  1  : : 
L  :  39.11,  or  L=39.11x36  inches=117i  feet. 

2.  Compare  the  times  of  vibration  T,  t,  of  two  pendulums 
whose  lengths  are  L,  /,  when  carried  to  the  distances  D,  d,  above 
the  Earth's  surface. 


142 


NATURAL   PHILOSOPHY. 


Let  7-^the  radius  of  the  Earth,  then  since  the  force  of  gravity 
varies  inversely  as  the  square  of  the  distance  from  the  Earth's  cen- 
ter, the  force  which  accelerates  the  pendulum  whose  length  is  (L)  : 

1 
the  force  which  accelerates  the  pendulum  whose  length  is  (I):  :  —  —  .« 

1         __  /L\* 

:==  ::  r+d\*  :  r+D|f;   but  by  Art  182,  T  oc  (^  )  f 


If  It—  I,  then  T  :  t  :  :  r+D  :  r-\-d  ;  i.  e.  the  times  of  vibration 
of  the  same  pendulum,  when  carried  to  different  heights  above 
the  Earth's  surface,  are  to  each  other  as  the  distances  of  those 
heights  from  its  center. 

3.  If  a  pendulum  at  the  Earth's  surface  vibrates  (rri)  times  in 
T"  ;  how  must  its  length  be  altered  so  that  it  may  vibrate  (n) 
times  in  T"  ? 

Let  L=the  length  of  the  pendulum  which  vibrates  (m)  times 
in  T",  and  L+#=the  length  of  that  which  vibrates  (n)  times  in 

T"  ;  then  by  Art.  183,  (since  F  is  given)  m  :  n  :  :  —=•  :  * 

\/L     V(L+x) 

.-.  m'  :  ?i*  :  :  L+x  :  L,  or  n*L+n>x=m*L,  and  g=K~ffl. 

n 

Let  m=n±y,  where  y  is  very  small  with  respect  to  n  ;  then 
(m'-tt')L    (±2ny+y')L    , 
—  -rji  —  »         —(rejecting  y*  as  very  small  with  respect 

to  2m/)-  —  *.  and  consequently  L+#=L-  —  -,  which  furnishes  us 

with  a  convenient  theorem  for  ascertaining  the  quantity,  by 
which  the  pendulum  of  a  clock  must  be  lengthened  or  shortened 
according  as  it  gains  or  loses  a  few  seconds  or  minutes  in  a  day.* 

*  Suppose,  for  instance,  that  a  clock  gains  3  minutes  in  a  day,  i.  e.  instead  of  per- 


forming  24x60x60  or  86400  vibrations  in  a  day,  it  performs  86400+180  or  86580 
vibrations  in  that  time  ;  then  n=86400  and  y=180,  .-.  -  ^=  =—  -  ;  and  although 


this  acceleration  of  the  pendulum  indicates  that  its  length  is  a  little  less  than  39.11 
inches,  yet  it  is  evident  that  in  finding  the  value  of  —  ,  L  may  be  assumed  equal  to 

39.11  inches  without  any  material  error  ;  hence  ^—  -  or      '      (=.16  of  an  inch)  is  the 

quantity  by  which  the  pendulum  must  be  lengthened  to  make  it  vibrate  seconds.  In 
the  example  here  given,  m  (=86580)  is  greater  than  n,  and  therefore  y  is  positive  ; 
if  the  pendulum  loses  a  certain  number  of  seconds  in  a  day,  then  m  is  less  than  n, 

and  consequently  y  is  negative  ;  the  value  of  —  -  must  in  this  case  be  subtracted 


MECHANICS.  143 

4.  A  pendulum,  which  vibrated  seconds  and  kept  true  time  at 
the  Earth's  surface,  was  carried  to  the  top  of  a  mountain,  and 
there  lost  (t)  seconds  in  an  hour  :  What  was  the  height  of  the 
mountain  ? 

Let  r=radius  of  the  earth,  #=height  of  the  mountain,  T=the 
number  of  seconds  in  an  hour  or  the  number  of  vibrations  in  an 
hour  at  the  Earth's  surface,  then  T—  t  will  be  the  number  of  vi- 
brations in  an  hour  at  the  top  of  the  mountain.  By  Art.  183, 
when  the  length  of  the  pendulum  is  given,  the  number  of  vibra- 
tions in  a  given  time  are  directly  as  the  square  roots  of  the  forces 
which  act  upon  the  pendulum  ;  hence,  since  the  forces  are  in- 
versely as  the  squares  of  the  distances  from  the  center  of  the  earth 

1.-L  - 

r      r+x' 


185.  QUESTIONS  ON  THE  PENDULUM. 

1.  The  lengths  of  pendulums  vibrating  seconds  at  St.  Thomas 
near  the  equator,  at  New  York,  at  London,  and  at  Spitzbergen,  in 
lat.  80°,  have  been  ascertained  by  very  accurate  experiments  to 
be  as  stated  below:  Required  the  space  through  which  a  body 
would  fall  in  one  second  at  those  places  respectively?  (Art.  181.) 

Length  of  Pend.  Space  in  1  sec. 

St.  Thomas,  39.02074  192.56  inches. 

New  York,  39.10168  192.96 

London,  39.13860  193.14 

Spitzbergen,  39.21469  193.52.f 

2.  What  is  the  length  of  the  seconds  pendulum  at  New  Haven, 
where  a  body  falls  from  rest  16-J-j  feet  the  first  second  ? 

Ans.  39.110,  or  39TV  inches  nearly. 

from  L,  or  the  pendulum  must  be  shortened.  The  theorem  alluded  to  in  the  3d  ex- 
ample, therefore,  gives  this  rule  :  Multiply  twice  the  length  of  the  pendulum  by  the 
number  of  seconds  gained  or  lost,  and  divide  the  result  by  the  number  of  seconds  in 
a  day  ;  the  quotient  will  give  the  number  of  inches  or  parts  of  an  inch  by  which  the 
pendulum  is  to  be  lengthened  or  shortened. 

*  If  t  be  very  small  with  respect  to  T,  then  x  may  be  considered  as  equal  to  = 
without  any  material  error.  Now  T=3600//,  if  therefore  t  be  equal  to  1,  2,  3,  &c. 
seconds,  then  x  will  be  equal  to  ,  ,  ,  &c.  ;  and  supposing  r  to  be  equal 


to  4000  miles,  the  heights  of  mountains  upon  which  a  pendulum,  that  vibrates  sec- 
onds at  the  Earth's  surface,  loses  1,  2,  3,  &c.  seconds  in  an  hour,  will  be  if,  2f,  3$, 
&c.  miles  respectively. 

t  Hence  it  appears  that  the  space  through  which  a  body  falls  from  a  state  of  rest 
in  one  second,  is  less  at  the  equator  than  at  the  latitude  of  80°  by  iVo1  inch,  or  nearly 
linch. 


144  .  NATURAL   PHILOSOPHY. 

3.  The  length  of  the  seconds  pendulum  being  39T'ff  inches, 
what  are  the  lengths  of  pendulums  vibrating  £  and  £  seconds ; 
also  in  2  seconds ;  and  how  long  must  a  pendulum  be  to  vibrate 
once  an  hour? 

Ans.    For  half  sec.  9.775  inch.= £  the  length  of  the  seconds  pend. 
quarter     "     2.444     "    =TV          " 
two  "   13.03    feet  =4 times"  " 

one  hour  7997.7     miles  =diameter  of  the  earth  nearly.* 

4.  A  pendulum  which  vibrated  seconds  at  the  level  of  the  sea, 
was  found  to  vibrate  but  3597  times  in  an  hour,  on  the  top  of  a 
neighboring  mountain :  Required  the  height  of  the  mountain  ? 

Ans.  3£  miles. 
. 

185''.  CENTRAL  FORCES. 

Central  Force  is  that  power  or  energy  with  which  a  body 
moving  in  any  curve  around  a  center  is  made  to  approach  to,  or 
recede  from  that  center.  In  the  former  case  it  is  called  the  cen- 
tripetal,  and  in  the  latter  the  centrifugal,  force. 

(1.)  When   a  body  revolves  in   a  Fig.  US'. 

circle  the  centripetal  and  centrifugal 
forces  are  equal  to  each  other.  In  Fig. 
118',  Eb  represents  the  force  by  which 
a  body  revolving  in  a  circle  around  the 
center  E,  is  carried  toward  the  center 
by  the  centripetal  force  ;  but  since, 
were  its  motion  not  thus  constrained  it 
would  have  receded  from  the  center 
through  the  space  B&,  consequently, 
this  line  also  represents  the  centrifugal 
force.  Hence  the  two  forces  are  equal 
to  each  other. 

(2.)  The  central  force  (either  the  centripetal  or  the  centrifugal) 
of  a  body  moving  uniformly  in  a  circle,  is  as  the  square  of  the  ve- 
locity divided  by  the  radius  of  the  circle.  Let  r  be  the  radius  of 
the  circle,  v  the  velocity  of  the  body,  and  Ab  the  arc  described 
in  an  indefinitely  short  time  t.  Then  (since  by  our  hypothe- 
sis the  motion  is  considered  as  uniform)  the  space  Ab=vt. 
But  Ab  being  a  very  small  arc,  may  be  taken  as  equal  to  its 
chord,  which  is  a  mean  proportional  between  the.  diameter  2r 
and  the  versed  sine  Aa,  (=B6).  Hence,  Aax2r=Ab*,  and  Aa= 

•—-=—,  which,  when  the  time  is  given,  varies  as  — » .\P  oc  — . 
2r      2r  r  r 

(3.)  Again,  the  central  force  is  as  the  radius  of  the  circle  divided 
by  the  square  of  the  time.  Let  t  equal  the  time  of  describing  the 

*  The  diameter  of  the  earth  is  7912  miles. 


MECHANICS.  145 

2<rr       r 
whole  circumference  =2#r ;    then  2«r=2u,  and  v=— — ,  oc  -  and 

us  oc  p.     But  F  x  -  x  ^oc  p,  .-.Foe  p. 


CHAPTER  X. 

OF  PROJECTILES. 

186.  A  body  thrown  into  the  air  at  any  angle  with  the  horizon,  is 
called  a  projectile. 

The  doctrine  of  Projectiles  proceeds  on  the  supposition  that  the 
force  of  gravity  acts  uniformly,  and  that  bodies  move  without 
resistance  from  the  air,  neither  of  which  suppositions  is  strictly 
true  ;  but,  for  distances  so  small  as  those  usually  involved  in 
these  inquiries,  the  variation  in  the  force  of  gravity  is  not  mate- 
rial, and  for  all  practical  purposes  the  effect  of  the  resistance  of 
the  air  is  separately  computed,  and  allowed  for. 

187.  If  a  body  be  projected' in  any  direction,  not  perpendicular 
to  the  horizon,  it  will  describe  a  parabola. 

This  proposition  has  already  been  demonstrated  in  Art.  48. 

The  most  concise  and  comprehensive  method  of  treating  this 
subject,  is  first,  to  investigate  by  means  of  Analytical  Trigonom- 
etry, a  general  formula,  showing  the  relations  between  the  time, 
velocity,  range,  and  angle  of  elevation  of  a  projectile,  however 
these  may  vary  among  themselves ;  and  secondly,  to  deduce  from 
this,  formulae  of  greater  simplicity  that  serve  for  particular  cases. 

188.  In  order  to  investigate  the  general  formula,*  let  A  (Fig. 
119,)  be  the  point  of  projection,  AB  or  AB'  the  plane  on  which 
the  body  is  projected,  passing  through  A.     AB  also  denotes  the 
range,  or  distance  to  which  the  body  is  thrown.     Let  AC  be 
drawn  parallel,  and  BCD  perpendicular  to  the  horizon  ;  let  the 
angle  of  elevation  CAD=a,  the  angle  of  depression  of  the  plane 
CAB=fr,  the  velocity  of  projection=v,  the  time  of  flight=/,  the 
range  AB=r,  and  the  space  fallen  through  by  gravity  in  one  sec- 
ond or  1GTV  feet=m. 

Then,  by  the  laws  of  uniform  motion,  at  the  end  of  the  time  t, 
if  gravity  did  not  act,  the  body  would  be  found  in  the  point  D, 

*  See  Encyc.  Metropolitan,  Art.  Mechanics,  p.  105,  from  which  a  large  part  of 
this  chapter  is  taken. 

19 


146 


NATURAL    PHILOSOPHY. 


while,  by  the  laws  of  falling  bodies,  it  would  in  the  same  time 
pass  through  the  perpendicular  DB,  consequently, 
AD=tv ;  and  DB=mr. 


Fig.  119. 


In  the  right-angled  triangles  ABC  and  ADC,  the  angle  B  is 
the  complement  of  b,  and  the  angle  D  is  the  complement  of  a  ; 
and,  since  the  sides  are  as  the  sines  of  the  opposite  angles, 

cos.   b  :  sin.  (adbft)*  ::  tv  :  fa8"1'  (a± 

cos.  b 


v          cos.  b 

.    ,  ,.           r  sin.  (a±b) 
Again,  cos.  a  :  sin.  (a±b)  ::r  : S - 


~    m?  _  sin.  (adb 
r  cos.  a 


cos.  a 


_,  „  x        v  sin.  (adb&) 

From  equation  (1.)  t  =  -  -—r^- 
m  cos.  b 


m  cos.   b 

r  sin.  (adbft) 

m  cos.  a 


(2.) 


From  equa.   (2.)   t*  = 

T,  r     sin.  (a ±b)  cos.  a 

Hence'V=       »J.-t    •  8-> 

From  these  three  equations,  all  the  relations  between  the  time. 


*  Plus  when  the  plane  descends,  minus  when  the  plane  ascends. 


MECHANICS.  147 

velocity,  range,  and  angle  of  elevation,  are  readily  determined  ; 
so  that  any  two  of  these  four  quantities  being  given,  the  other 
two  may  be  found.  Thus, 

mt  cos.  b 
By  equation  (l.)i;=sin(a±6). 

mt*  cos.  a 
By  equation  (2.)  r=^-(^- 

If  the  range  and  elevation  be  giveri  to  find  the  time  and  velocity, 

/rsin.  (a±&)\* 

By  equation  (2.)  t  =  I  -  ^  -  •  1   • 
J    ^  \    m  cos.  a    / 


(       rm  cos*b      \ 

By  equation  (3.)  v  =[  -T—  -  -  -  --  I   • 
\sm.(a±6)  cos.  «/ 

If  the  velocity  and  elevation  be  given  to  find  the  time  and  range 

„  „  A  :.    f  sin.  («±6) 

By  equation  (1.)  t  =  -  -  —  r~. 
m  cos.  6 

v9  sin.  (a  ±6)  cos.  a 
By  equation  (3.)  r  =  -  ^TJ~ 

If  any  two  of  the  above  quantities  be  given  to  find  the  angle 
of  elevation,  then  (the  inclination  of  the  plane  to  the  horizon  (6) 
being  supposed  to  be  known)  in  order  to  find  the  value  of  a,  we 
must  substitute  for  sin.  (a±&),  its  value  involving  the  sine  of 
each  are  separately.  Now  sin.  (a  ±b)  —sin.  a  cos.  6±sin.  b  cos. 
a,*  whence  either  the  sin.  a  or  cos.  a  may  be  obtained. 

The  value  of  a  and  b  being  known,  we  shall  then  find  in  all 
the  preceding  cases,  sin.  (a±b)  cos.  a  equal  to  a  known  quantity. 
Let  this  be  denoted  by  C  ;  then  cos.  a  sin.  a  cos.  &±sin.  b  cos.sa=C  ; 

Let  a:=sin.  a,  then  cos.  «=(!—  <zs)J 

*(!-:£')*  ±tan.f  b(l-*S)=' 


189.  As  this  is  a  quadratic  equation,  the  solution  will  give  two 
values  of  x,  which  shows,  that  there  are  always  two  different 
angles  of  elevation,  which  equally  satisfy  the  conditions  of  the 
problem,  except  in  the  case  where  the  two  roots  of  the  equation 
are  equal  to  each  other,  when  only  one  angle  will  be  found.  In 
this  case,  as  is  shown  below,  sin.  a=sin.  £  (90  ±&),  under  which 
limitation  the  range  will  be  a  maximum,  or  the  greatest  possi- 
ble ;  and  all  angles  of  elevation  equally  above  and  below  that 
which  gives  the  maximum  range,  will  give  ranges  equal  to  each 
other.  For,  by  the  value  of  r  as  determined  above, 

sin.  (a±b)  cos.  a 
r=v  -  s-j  -  • 
m  cos.  b 

»  Day's  Trigonometrical  Analysis,  Art.  208.  t  Ib.  Art  216. 


148 


NATURAL    PHILOSOPHY. 


If  v  and  the  angle  b  are  given,  the  range  will  vary  as  sin. 
(a ±b)  cos.  a.     But 
sin.  (a±b)  cos.  a=sin.  a  cos.  a  cos.  &±sin.  6  cos.1  a. 

=£  sin.  2  a  cos.  6±sin.  6(1+1  cos.  2a)* 
=i  sin.  2  «  cos.  6±£  cos.  2  a  sin.  &-fi  sin.  (±6) 
=i  sin.(2  a±6)+i  sin. (±6)  ;  and  since  the  second 
part  of  this  expression  is  constant,  the  range  will  vary  as  sin. 
(2a±b).     This  quantity  will  be  greatest,  when 
1  2a±b=QO°. 

Then  2a±b=QO°±b 


sin.  a=sin.  i(90°±&). 

Therefore,  as  above,  the  range  will  be  a  maximum  where  the 
sine  of  the  angle  of  elevation  equals  the  sine  of  i(90° ±6). 

And  since  all  angles  equally  above  and  below  90°  have  the 
same  sine,  all  angles  equally  above  and  below  that  which  gives 
the  maximum  have  equal  ranges.  Thus,  a  cannon  ball  fired  at 
an  angle  of  60°  (in  a  vacuum)  above  a  horizontal  plane,  would 
reach  the  plane  at  the  same  distance  from  the  point  of  projection 
as  if  fired  at  an  elevation  of  30°.  When  the  data  of  the  problem 
give  or  require  a  greater  value  for  sin.  (2a±b)  than  1,  that  is, 
than  the  sine  of  90°,  it  s.  .  o  the  problem,  under  the  proposed 
conditions,  to  be  impossible. 

190.  To  find  the  greatest  height  to  which  the  projectile  will 
ascend,  we  must  recollect  that  a  body  projected  perpendicularly 
upward,  will  rise  to  the  same  height  from  which  it  must  have 
fallen  to  acquire  the  velocity  of  projection.  (Art.  30.)  Calling 
the  time  of  rising  to  the  greatest  height  t',  we  shall  have  2mt' 
=the  velocity  of  descent  from  gravity ;  and,  v  representing  the 
whole  velocity  of  projection  in  an  oblique  direction,  the  perpen- 
dicular part  will  be  represented  by  v  sin.  a  ;  whence 

,    .    v  sin.  a      ,   lCt    i>2  sin.2  a 

2mt'=v  sin.  a,  and  t'= and  t'z= -. — 

2m  4m2 

But  the  space  fallen  through  in  the  time  t'=mt'z= — £^L_^. 

4m 
And  the  ascent  in  the  same  time  from  projection  is, 

t'v  sin.  a  — — — - — .     Consequently,  the   difference  of  these 

will  be  the  greatest  height  of  the  projectile  above  the  point  A ; 

At  4  •     i.    «2sin.2a     t^sin.'a    v2sin.2a       .  4mh  ... 

that  is,  h= = ,  and  — 5-  =  sin.2  a.        (4.) 

2m  4m  4m  u2 

If  therefore  the  angle  of  elevation  and  velocity  are  given,  the 

*  Day's  Trigonometrical  Analysis,  Art.  213. 


MECHANICS.  149 

greatest  height  may  be  determined  ;  or  if  the  range  (r)  or  the 
time  (t)  be  given,  (the  angles  being  known,)  the  value  of  v1  may 
first  be  ascertained  by  preceding  formulae,  and  then  the  height  (h) 
from  equation  (4.) 

191.  All  the  preceding  equations  become  much  more  simple 
when  the  projection  is  along  a  horizontal  plane  ;  for  then  6=0, 
and  sin.  6=0,  and  cos.  6=1  ;  hence, 


T          /*\mt(     sin.(arb6)\       . 
In  eq.  (1.)—  I  =  -  *  —  ^  }  =sm.  a. 
v  \         cos.  6    / 


a=tan.  a.  (2'.) 


r   \         cos.  a  cos.  a 

r  /sin.  (a±6)  cos.  a\  _  sin.  a  cos.  a  _  %  sin.  2a 

~ 


n          f        /-./  N        vxsin.  a  ,„  . 

Hence  from  (I'.)  £=  -  oo  uxsm.  a.  (5.) 

(6.) 


.  (7.) 


On  a  horizontal  plane,  therefore,  (the  most  usual  case,)  we  have 
the  following  THEOREMS. 

I.  The  TIME  OF  FLIGHT  varies  as  the  velocity  of  projection  multi- 
plied by  the  sine  of  the  angle  of  elevation. 

II.  The  RANGE*  varies  as  the  square  of  the  velocity  of  projection, 
multiplied  by  the  sine  of  twice  the  angle  of  elevation. 

III.  The  GREATEST  HEIGHT  varies  as  the  square  of  the  velocity  of 
projection,  multiplied  by  the  square  of  the  sine  of  the  angle  of  ele- 
vation. 

Moreover,  since  the  sine  of  twice  45°  equals  the  sine  of  90°, 
which  equals  radius,  hence,  by  Theorem  II, 

IV.  The  RANGE  is  GREATEST  when  the  angle  of  elevation  is  45°, 
and  is  the  same  at  elevations  equally  above  and  below  45°.f 

And  since  the  square  of  the  sine  of  the  angle  of  elevation 
must  be  greatest  when  the  angle  is  a  right  angle,  therefore,  by 
Theorem  III, 

V.  A  projectile  rises  to  the  GREATEST  HEIGHT  when  thrown  perpen- 
dicularly upward. 

Finally,  since  the  sine  of  the  angle  of  elevation  is  greatest 
when  the  angle  is  a  right  angle,  therefore,  by  Theorem  I, 

VI.  The  TIME  OF  FLIGHT  is  GREATEST,  when  the  body  is  thrown  per- 
pendicularly upward. 

*  Sometimes  called  random. 

t  For  the  sine  of  twice  any  angle  below  45°  is  the  same  as  the  sine  of  twice  any 
angle  of  the  same  number  of  degrees  above  45°. 


150  NATURAL  PHILOSOPHY. 

192.  QUESTIONS  ON  PROJECTILES. 

1.  A  body  is  projected  at  an  angle  of  15  degrees  with  the  hori- 
zon, with  the  velocity  of  140  feet  per  second  :  What  is  its  range, 
greatest  height,  and  time  of  flight  ? 


By  (6.)  r=  =  =304.663  feet. 


By  (4.)  h=  =  (by  logarithms)  20.409  feet. 

By  (5.)  f=VXSI"'g  =2.253  seconds. 
in 

That  is,  its  horizontal    range  is   304.663  feet  ;    its  greatest 
altitude  20.409  feet  ;  and  its  time  of  flight  is  2.253  seconds. 

2.  A  body  was  projected  at  an  angle  of  60°  with  the  horizon,  and 
descends  to  it  at  the  distance  of  100  feet  :  With  what  velocity 
was  it  projected,  and  what  was  its  greatest  altitude  and  its  time 
of  flight  ? 

From  (6.)  v=  (  -^-\  =60.94  feet. 
Vsm.  2a/ 

From  (4.)  4=^1^=43.3  feet. 

4m 

From  (5.)  t=VXSm'a=3.28  seconds. 

3.  I  fired  an  arrow  which  remained  in  the  air  4  seconds,  and 
fell  at  the  distance  of  100  feet  :  With  what  angle  of  elevation 
was  it  fired,  with  what  velocity,  and  how  high  did  it  ascend  ? 

By  equation  (2'.)  tan.  «=—  =2.57333=tan.  68°  46'. 
By  equation  (6.)  v=(\  =69.024  feet  per  second. 


By  equation  (4.)  h=    _l= 

4.  A  gun  was  fired  at  an  elevation  of  50°,  and  the  shot  struck 
the  ground  at  the  distance  of  4898  feet  :  With  what  velocity  did 
it  leave  the  gun,  and  how  long  was  it  in  the  air  ? 

Ans.  Velocity,  400  feet  per  second. 
Time,  19.05  seconds. 

5.  Random  4898  feet,  time  of  flight  16  seconds  :  Required  the 
angle  of  elevation  and  the  velocity  of  projection  ? 

Ans.  El.  40°  3',  V.  400  feet  per  sec. 

6.  Random  2898  feet,  velocity  of  projection  389.1  feet  :  What 
were  the  elevation,  and  time  of  flight  ? 

Ans.  EL  19°  or  71°,  T.7.87  -<;c. 


MECHANICS.  151 

7.  Elevation  40°,  random  4898 :  Required  the  random  when 
the  elevation  is  29|°  ?     Art.  191.  (6.)  Ans.  4263. 

8.  Elevation  40°  3',  time  of  flight  16  seconds :  Required  the 
random  and  velocity  of  projection  ? 

Ans.  R.  4898,  V.  400  feet. 

9.  Velocity  510  feet  per  sec.,  time  of  flight   15  seconds,  to 
find  the  elevation  and  random.  Ans.  El.  28°  14',  R.  6740. 

10.  On  a  side-hill  ascending  uniformly  above  a  horizontal  level 
at  an  angle  of  10°  20',  a  ball  was  fired  at  an  angle  of  elevation 
above  the  horizon  of  34°,  and  with  a  velocity  of  401  feet  per 
second  :  What  was  the  range  on  the  hill-side  when  the  gun  was 
directed  up  the  hill,  and  what  when  directed  downward  ? 

Ans.  3438  and  5985  feet.* 


CHAPTER  XL 

OF  THE  STRENGTH  OF  MATERIALS. 

193.  THE  importance  to  the  architect  and  the  engineer  of  as- 
certaining the  form  and  position  of  the  materials  which  he  em- 
ploys, in  order  to  secure  the  greatest  degree  of  strength  and 
stability   at   the   least   expense,   has   led  mathematicians  and 
writers  on  mechanics,  to  devote  much  attention  to  this  subject. 
How  is  the  strength  of  a  beam  affected  by  giving  to  it  different 
shapes  and  different  positions  1  how  must  a  given  quantity  of 
matter  be  disposed  in  order  that  it  may  have  the  greatest  possi- 
ble strength  ?  and  upon  what  principles  depends  the  stability  of 
columns,  roofs,  and  arches  ?  these,  and  many  similar  inquiries, 
have  been  the  objects  of  profound  investigation. 

The  strength  of  beams,  or  pieces  of  timber,  is  the  first  object 
of  inquiry.  STRENGTH  is  the  power  to  resist  fracture  :  STRESS,  the 
power  to  produce  fracture. 

194.  The  strength  of  a  beam  resting,  horizontally,  on  its  two  ends, 
from  a  weight  on  its  center,  is  proportioned  to  the  area  of  a  cross 
section,  multiplied  by  the  depth  of  its  center  of  gravity. 

Thus,  in  Fig.  120,  if  AB  represent  a  stick  of  timber,  resting 
horizontally  on  supports  at  its  two  ends,  and  W  be  a  weight 
placed  at  the  center,  and  a  b  c  d  be  a  cross  section,  then,  (sup- 
posing the  weight  to  be  sufficient  to  break  the  beam,)  the  frac- 
ture will  commence  at  the  bottom,  and  proceed  regularly  to  the 

«  See  Art.  188,  formula  (3.) 


152 


NATURAL   PHILOSOPHY. 


top,  ending  at  W.*  Now  since  the  tendency  to  resist  fracture 
from  cohesion,  must  depend  upon  the  total  amount  of  all  the  sep- 
arate forces  acting  between  contiguous  particles,  it  must  evi- 
dently depend  upon  the  number  of  the  particles,  that  is,  upon 
the  extent  of  surface,  or  area  of  the  section. 

Fig.  120. 
A  W  B 


This  would  be  the  case  were  no  other  mechanical  force  in- 
volved but  cohesion ;  but  the  reaction  of  the  supports  at  A  and 
B,  being  equal  to  the  weight,  and  severally  acting  at  the  longer 
end  of  a  bent  lever  AWa,  AWb,  &c.,  consequently,  the  tendency 
to  fracture,  from  the  leverage,  will  be  lessened,  as  the  shorter 
arm  of  the  lever  is  increased,  while  the  longer  arm  remains  the 
same  :  therefore  the  strength  being  inversely  as  the  stress,  will 
be  regularly  increased  as  the  distance  of  any  lamina  from  W  is 
increased,  and  the  whole  effect  will  be  as  the  distance  of  the 
center  of  gravity  from  that  point.  Hence  from  both  causes,  the 
strength  varies  as  the  area  of  the  section  multiplied  by  the  depth 
of  the  center  of  gravity,  f 

Fig.  121. 
C  A 


195.  This  proposition  is  general,  and  applies  to  a  number  of 
distinct  cases.  In  cylindrical  and  square  beams,  since  the  area 
of  the  section  varies  as  the  square  of  its  diameter,  and  the  dis- 
tance of  the  center  of  gravity  from  the  point  E,  (Fig.  121,)  varies 
as  the  diameter,  their  strength  is  as  the  cubes  of  the  diameters. 
In  beams  of  an  oblong  figure,  the  strength  varies  as  the  breadth 
and  square  of  the  depth  ;  for  here  the  area  being  as  the  product 
of  the  two  sides,  and  the  distance  of  the  center  of  gravity  from 

*  It  is  here  supposed,  according  to  the  views  embraced  by  most  writers  on  Mechan- 
ics since  the  time  of  Galileo,  that  the  parts  of  a  fractured  beam  turn  about  the  line 
where  the  fracture  terminates  ;  but  Mr.  Barlow,  in  his  essay  on  the  Strength  and  Stress 
of  Timber,  proves  by  experiment,  that  the  tendency  is  to  turn  about  a  line  entirely 
within  the  section,  the  fibres  011  that  side  of  the  line  where  the  fracture  begins  being 
extended,  and  those  on  the  other  side  compressed.  This  line  he  calls  the  neutral  axis. 

t  Day's  Algebra,  (Art.  408.) 


MECHANICS. 


153 


E  being  equal  to  half  the  perpendicular  side,  and  therefore  pro- 
portioned to  that  side,  the  proposition  is,  that  the  strength  varies 
as  the  breadth  x  depth  x  depth,  or  as  the  breadth  into  the  square 
of  the  depth.  Hence,  the  same  oblong  beam  with  its  narrow 
side  upward,  is  as  much  stronger  than  with  its  broad  side  up- 
ward, as  the  depth  exceeds  the  breadth.  For  the  area  being 
the  same  in  both  cases,  the  strengths  are  proportioned  to  EG  and 
Eg1,  or  as  AB  to  AC.  Thus  if  a  joist  be  10  inches  broad  and  2|- 
thick,  it  will  bear  four  times  more  weight  when  laid  on  its  edge 
than  on  its  side.  Hence  the  modern  mode  of  flooring  with  very 
thin,  but  deep  pieces  of  timber. 

196.  In  beams  of  different  lengths,  resting  on  two  supports,  the 
strength  will  vary  as  the  area  of  the  section  into  the  depth  of  the 
center  of  gravity,  divided  by  the  length  into  the  weight. 

Let  L,  /,  denote  the  lengths  ;  W,  w,  the  weights  ;  A,  a,  the 
areas  of  the  sections  ;  and  G,  g,  the  depths  of  the  centers  of 
gravity,  of  two  prismatic  beams,  resting  horizontally  on  their 
two  ends. 

The  stress,  or  tendency  to  produce  fracture,  from  the  weight  of 
the  beam  itself,  will  be  expressed  by  |LxW,  and  ±lxw.*  But  the 
tendency  to  resist  fracture  is  denoted  by  AxG,  and  axg.  Hence 
the  aggregate  strength  of  the  timber  will  be  directly  as  the  latter 
and  inversely  as  the  former.  That  is, 

AxG       axg      AxG    axg 
:  *  :  :  :  :  :  LxW  :  Txw 


197.  A.  triangular  beam  is  twice  as  strong  when  resting  on  its 
broad  base,  as  when  resting  on  its  edge. 

For,  the  area  being  the  same  in  both  cases,  the  strength  varies 
as  EG  and  Eg,  (Fig.  122,)  which  are  as  2  to  1.     (Art.  71.) 

Fig.  122. 
E  A  E  B 


These  principles  apply  not  only  to  beams,  but  to  bars,  and 
similar  structures  of  every  sort  of  matter. 

198.  The  strength  of  any  bar  in  the  direction  of  its  length,  is 
directly  proportional  to  the  area  of  its  transverse  section. 

*  For  the  reaction  at  each  support  is  a  force  which=JW,  acting  upon  £L  at  its 
center  of  gravity  ;  but  the  center  of  gravity  of  $L,  is  at  a  distance  from  the  prop  =  J 
of  JL  =  JL  ;  therefore  the  efficacy  of  the  force  =  iLxiW  =  iLxW.  (Fig.  120.) 

20 


154  NATURAL    PHILOSOPHY 

Here  each  line  of  particles,  in  a  longitudinal  direction,  may  bo 
considered  as  exerting  a  separate  force,  and  therefore  the  aggre- 
gate force  will  manifestly  depend  on  their  number,  and  of  course 
on  the  area  of  its  section,  the  whole  being  equal  to  the  sum  of 
all  its  parts.  Hence  the  various  shapes  of  bars  make  no  differ- 
ence in  their  absolute  strength,  since  this  depends  only  on  the 
area  of  the  section,  and  must  obviously  be  the  same  when  the 
area  is  the  same,  whatever  be  its  figure.  A  rope,  therefore,  or  a 
wire,  to  which  a  weight  is  appended,  is  as  likely  to  break  in  one 
place  as  in  another,  but  when  the  weight  of  the  rope  becomes 
considerable,  and  the  force  is  applied  perpendicularly,  the  in- 
crease of  weight,  as  its  length  increases,  renders  it  more  liable 
to  break  in  the  upper  than  in  the  lower  parts. 

199.  The  lateral  strengths  of  similar  beams,  are  inversely  as 
their  lengths  or  breadths. 

Let  D,  d,  represent  the  diameters  of  two  cylindrical  beams,  or 
the  sides  of  two  beams  in  the  form  of  square  prisms  ;  then, 

D3          T3 

S  :  s  :  :  =  —  ==  :  j  --    In  similar  beams,  L  oc  D  and  W  oc  D3 
LxW  Ixw 

Q          D3  d3      I    1      1    1 
::D*:d4::D:d::LT 
In  oblong  beams, 

BxD*    bxd*    „  _       _ 

-=  —  ™  :  -j  -  •    But  in  similar  beams,  L  oc  B, 

LxD8    Ixd*     _!_    1     !_    1 
::  ''  lX(Pi:D:d'':L:  /' 


When  the  beams  are  of  the  same  figure,  and  their  lateral  sec- 
tions the  same,  then,  the  breadth  andvdepth  of  one  being  respect- 
ively equal  to  the  breadth  and  depth  of  the  other, 


:'    And  since 


Hence,  half  the  length  of  a  beam,  supported  at  both  ends,  will 
bear  four  times  as  great  a  pressure  as  the  whole  beam  ;  and  a 
prop  placed  under  the  center  of  a  beam  increases  its  strength  in 
the  same  ratio. 

Long  beams  are  weak  from  their  own  weight  ;  and  the  length 
may  be  so  increased,  that  they  will  break  from  this  cause  alone." 
The  strength  arising  from  making  the  beam  larger,  increases  as 
the  square  of  one  of  the  homologous  sides,  while  the  weight  in- 
creases as  the  cube,  and  therefore  preponderates  in  long  beams. 
To  consider,  in  connection,  the  several  circumstances  which  af- 
fect the  strength  of  timber,  it  appears  that  a  beam  twice  as 
broad  as  another  of  the  same  length,  is  also  twice  as  strong  ; 
that  one  twice  as  deep,  the  other  dimensions  remaining  the  same, 
is  four  times  as  strong  ;  and  that  one  twice  as  long  as  another 
similar  beam,  has  only  half  the  strength. 


MECHANICS.  155 

200.  But  if  besides  their  own  weights,  these  beams  are  made 
to  support  other  weights,  W,  w',  placed  at  their  middle  points, 
then  their  tendency  to  fracture  will  be  increased. 

For,  since  the  reaction,  arising  from  the  pressure  on  each  sup- 
port^W',  and  this  force  acts  at  the  point  of  fracture  with  a 
leverage  equal  to  |L,  the  stress  produced  by  W=%Lx±W= 
iLxW.  But  the  stress  arising  from  the  weight  of  the  beam  it- 
self=}LxW.  Therefore  the  whole  stress=iLxW-f  |LxW' 
=*L(iW+W')  oc  L  (iW+W'), 

AxG 


or,  in  the  case  of  cylinders  and  square  prisms, 


''  L(iW+W) 

If  the  weights  of  the  beams  be  so  small,  when  compared  with 
the  weights  supported,  as  to  make  it  unnecessary  to  take  them 
into  consideration,  then 

Q  AxG      <*xg  D3          d3 

S:s::  '  :  :  :  <m  ^  and  s 


201.  In  order  that  the  foregoing  general  formulae  may  be  ap- 
plied to  practice,  so  as  to  find  the  actual  strength  of  bars  or  beams, 
it  is  necessary  to  have  some  standard  of  strength  ascertained  by 
experiment,  which  may  be  employed  as  the  unit  of  comparison. 
For  example,  it  is  found  by  experiment  that  a  small  beam  of  oak, 
one  foot  long  and  one  inch  square,  is  able,  when  supported  at  both 
ends,  to  sustain  a  weight  of  600  pounds  ;  and  that  a  bar  of  iron 
of  the  same  dimensions,  would  sustain  in  the  same  circumstances 
2190  pounds.  The  beam  weighs  half  a  pound,  and  the  iron  three 
pounds.  With  these  data  applied  to  the  foregoing  formulae,  we 
may  perform  such  problems  as  the  following. 

1.  What  weight  might  be  sustained  at  the  middle  point  of  a 
prismatic  beam  of  oak,  whose  length  is  6  feet,  and  its  end  4  inches 
square  ? 
Let  S=strength  of  the  beam  required. 

s=strength  of  a  beam  whose  length  is  one  foot  and  square 
end  one  inch. 

W=weight  of  the  larger  beam,  and  w  that  of  the  smaller=i 
pound.  Let  L=6,  /=!,  D=4,  d=I.  Weight  required=W'.  Giv- 
en weight  (600  pounds)  =«/. 

Then,  the  weight  of  the  beams  not  being  taken  into  the  ac- 

D3  d3  4*  1s 

count,  S  :  ,  :  :  j-^  :  ^  :  :  —^,  :  ^^. 

But  the  strength,  at  the  moment  of  fracture  =0  in  both  cases,  L  e. 
S=Si  •••=i  whence  W'=6400  pounds. 


156  NATURAL   PHILOSOPHY. 

If  the  weight  of  the  beams  be  taken  into  the  account,  then  (Art, 

200.)  8 :.::  -^E=  :  -^=-  ::_=£ 
LxiW+W'      Ixfa+wf     6X24*+ 


1  xi+600 
64  1 

and  W=6378f  pounds.     Ans. 

2.  What  must  be  the  depth  of  a  beam  of  an  oblong  prismatic 
form,  whose  breadth  is  2  inches  and  length  8  feet,  to  support  a 
weight  of  6400  pounds,  its  own  weight  not  taken  into  consid- 
eration ? 

Here,  (Art. 

85.333,  or  D=6.53  inches.     Ans. 

3.  What  weight  might  be  supported  at  the  middle  point  of  a 
bar  of  iron  10  feet  long,  and  the  side  of  whose  square  end  is  3 
inches,  its  own  weight  not  being  taken  into  consideration  ? 

Ans.  5913  pounds. 

202.  The  stress  (or  tendency  to  fracture)  on  any  part  of  a  hor- 
izontal beam  supported  at  both  ends,  is  proportional  to  the  product 
of  its  two  distances  from  the  supported  ends. 

Fig.  123.  > 

AC8  B 


W 


The  sum  of  the  pressures  on  A  and  B,  (Fig.  123,)  must  obvi- 
ously be  equal  to  the  whole  weight.     But,  (Art.  102,) 

~  WxBC        ,  _    WxAC 

Jrressure  at  A=  —  .        ,  and  pressure  at  tt=  —  -pg  —  . 

Ai>  AiJ 

But  the  reaction  of  either  point  of  support  is  equal  to  the  pres- 
sure on  that  point  ;  and  this  force  acts  at  C  with  a  leverage  AC  on 

~ 


one  side,  and  BC  on  the  other,  so  that  the  stress  at  C=  —  irFr- 

Ari 

WxAC 
xAC  or  —  TT>~  XBC,  either  of  which  expressions—  stress  at  C, 

A.D 

and  x  ACxBC.  And  since  this  rectangle  is  greatest  when 
AC=CB,  and  diminishes  as  these  lines  become  more  and  more 
unequal  in  length,  so  the  tendency  of  a  horizontal  bar  to  break 
is  greatest  in  the  middle,  and  decreases  toward  the  points  of 
support. 

•  For  W  :  to  :  :  LxD»  :  W,  .-.W  :  *  :  :  6x16  :  1  :  :  96  :  1,  .-.W=48. 


MECHANICS. 


157 


Hence  a  beam,  in  order  to  be  equally  strong  throughout,  must 
be  thickest  in  the  middle,  being  thinned  off  toward  the  ends ; 
and  if  the  sides  of  such  a  beam  are  parallel  planes,  the  figure  of 
the  beam  must  be  elliptical. 

Fig.  124. 
P  F 


For  let  the  curve  APDM,  (Fig.  124,)  whose  axis  is  AD,  repre- 
sent this  longitudinal  section,  and  let  a=ihe  thickness  or  breadth 
of  the  beam  ;  then  a  lateral  section  of  the  beam  at  any  point  C, 
will  be  an  oblong,  whose  breadth  is  a,  depth  PM,  and  the  depth 
of  its  center  of  gravity  £PM.  Hence,  the  tendency  of  the  beam 
to  resist  fracture  at  any  point  C,  is  as  axPM2  ;  but  the  stress  at 
C  is  as  ACxCD  ;  therefore 

,     4«     0xPM2        PM2 
the  strength  at  C*  —- 


hence  if  PM2  oc  ACxCD,  the  strength  will  be  the  same  at  every 
point  :  but  in  this  case  the  curve  APDM  is  an  ellipse,  whose 
transverse  is  AD,  and  conjugate  FK. 

203.  The  timbers  which  compose  the  horizontal  part  of  the 
frame  of  a  house,  being  usually  rectangular  parallelepipeds  of 
uniform  dimensions  throughout,  it  is  manifest  that  a  considerable 
portion  of  the  material  is  wasted  ;  but,  in  such  cases,  the  attempt 
to  save  the  material  would  be  attended  with  paramount  disad- 
vantages. When  however  the  material  is  expensive,  or  where 
lightness  is  important,  as  in  many  kinds  of  machinery,  the  fore- 
going principles  may  be  applied  with  great  advantage.  A  use- 
ful application  of  it  is  seen  in  the  shape  sometimes  given  to  the 
iron  bars  of  railways,  as  represented  in  the  following  figure. 
Fig.  125. 


204.  EXAMPLES. 

1.  What  must  be  the  length  of  a  beam  4  inches  square,  to 
support  6400  pounds  at  its  middle  point  ? 
Let  S=strength  of  the  required  beam, 

s=strength  of  a  beam  1  foot  long,  and  its  end  1  inch  square. 


158  NATURAL   PHILOSOPHY. 

W=weight  of  the  larger  beam, 
w=weight  of  the  smaller  beam  or  £  Ib. 
By  the  question,  L=required  length,  1=1,  D=4,  d—1. 
W'^6400  Ibs. 
M/=weight  sustained  by  the  unit=600  Ibs. 

(1.)  If  the  weight  of  the  beams  be  not  taken  into  considera- 
tion, by  Art.  200, 

g  .  g  . .     D3     ;    <P_ . . 43  _  .      I3 

Lx  W  '  lxwr ' '  Lx6400  '  1x600* 

But,  at  the  moment  of  fracture,  the  strength  in  both  cases  be- 
comes equal  to  nothing  ;  and  then,  S  being  equal  to  s,  ? = 

JL  X  6400 

whence  L— 6  feet. 


1X600' 
(2.)  If  the  weight  of  the  beams  be  taken  into  account,  then,  (Art. 

200)  s-*-- 5!   -•_*_-•. £_-. !L_ 

' '  L(*W+W)  '  /(iw+io')  ' '  L(4*+6400)  '  1  (1+600) 
64  1 

•'•E(iL+6400)=T+600-    Whence  L=5.98  feet. 

2.  What  must  be  the  breadth  of  a  beam  of  an  oblong  pris- 
matic form,  whose  depth  is  8  inches  and  length  6  feet,  to  support 
a  weight  of  6400  Ibs.,  its  own  weight  not  being  taken  into  the 
account  ? 

Let  B=required  breadth. 

6=breadth  of  the  beam  1  foot  long,  and  its  end  one  inch 

square. 

Then,  (Art.  200,) 

Q  BxD2     b3         Bx64         I3        __ 

S:S::I^^:J^'::6-^6400:T^600-    Hence  at  the  moment 

of  fracture,  wfen  S=s,  6xg4QQ  =^QQ'  whence  B=l    inch.     In 

the  two  preceding  examples,  the  beams  are  of  the  same  length, 
and  have  equal  strengths,  each  supporting  a  weight  of  6400  Ibs. 
But  in  beams  of  equal  length,  the  solid  contents  are  as  the  areas 
of  the  sections.  In  example  1st,  the  section=42=16 ;  and  in  ex- 
ample 2d,  the  section=l  x8=8.  Hence, 

The  first  beam  :  second  beam  : :  16  :  8.  Therefore,  the  oblong 
beam  placed  edgeways  is  as  strong  as  the  square  one,  although 
it  contains  only  one  half  as  much  material. 

3.  What  weight  may  be  supported  at  the  middle  of  a  bar  of 

*  The  weights  being  as  the  solid  contents, 

W  :  to  : :  LxDa :  Ixd*  : :  Lxl6  :  iXl,  .'.W  :  i  : :  Lxl6  :  1,  .-.iW=XL4 


MECHANICS.  159 

iron,  10  feet  long,  and  the  side  of  whose  square  is  3  inches,  its 
own  weight  being  taken  into  the  account  ? 

D3  6?       .  .  _  3_3  _  I3 

''  ''  10(135*+W)  *1(H+2190)* 


Hence,  W'=5782.05  Ibs. 

205.  The  foregoing  investigations  and  examples  relate  to 
beams  supported  at  both  ends  :  we  proceed  to  the  case  where 
the  beam  is  supported  at  only  one  end. 

In  similar  cylindrical  and  prismatic  beams,  supported  at  one 
end,  the  strength  varies  inversely  either  as  the  diameter  or  as  the 
length. 

Let  ABEF,  abef,  (Fig.  126,)  re- 
present the  longitudinal  sections  of 
two  prismatic  beams  fixed  horizon- 
tally into  the  wall  HKLM  ;  then  the 
tendency  of  these  beams  to  resist 
fracture  at  the  ends  EF,  ef,  where 
they  are  inserted  into  the  wall,  will 
be  measured  in  the  same  manner  as 
in  the  preceding  cases,  that  is,  by  the 
area  of  the  lateral  section  into  the 
depth  of  its  center  of  gravity,  except 
that  in  this  case,  the  fracture  will 
begin  at  the  upper  points  F,f,  and 
end  at  the  lower  points  E,  e.  The 
tendency  to  produce  fracture  will  be  the  weight  of  the  beams 
acting  at  the  distance  of  their  centers  of  gravity  from  the  ends 
EF,  ef.  Hence,  (Art.  196,) 

S  :  s  :  :  —  —  —  :       °  ,  or  if  any  weights,  W,  u/,  are  placed  at 

•jLj  X  W       ^IXw 

the  fother  ends  of  the  beams,  then  (since  the  effects  of  these 
weights  to  produce  fracture  will  be  measured  by  W'xL  and 
AxG 


we  have  S  :  .  :  :  -===  :  _=;  and  if  the  weights 
W,  10  of  the  beams  are  very  small  when  compared  with  the 
weights  W  Athens  :. 


Hence,  in  similar  beams,  as  in  Art.  199,  3  :*::—:-  or  —:-. 

Da      LI 

Let  W,  w,  represent  the  weights  of  the  parts  ABCD,  abed,  of 
the  beams,  then  the  tendency  of  those  parts  to  produce  fracture 
at  C,  c,  will  be  measured  by  iACxW,  and  ±acxw  ;  therefore,  if 


*  The  bar  of  iron  weighs  270  Ibs. 


160  NATURAL    PHILOSOPHY. 

S,  *  represent  the  strengths  of  the  beams  at  the  points  C,  c,  then 
AxG  ax 

;  or  if  w'  "•  be  veir  sma11 


with  respect  to  W  W,  then  S  :  ,  :  : 

Hence  if  a  given  weight  W'  be  supported  at  the  end  of  a  given 
beam  whose  weight  is  so  small  as  not  to  be  taken  into  considera- 
tion, the  strength  of  that  beam  to  support  the  weight  W  at  any 

point  C,  between  A  and  F,  will  vary  as  AQXTV,  5  or  since  W  is 

AxG 

constant,  as  _—. 
AO 

A  beam  supported  at  one  end  in  the  form  of  an  ISOSCELES  WEDGE, 
or  of  a  PARABOLA,  is  equally  strong  throughout  ;  or,  when  a  weight 
is  hung  at  the  end,  the  beam  is  as  liable  to  break  in  one  place  as  in 
another. 

Let  the  beam  be  in  the  form  of  an  isosceles  wedge,  (Fig.  127,) 
whose  flat  sides  are  parallel  to  the  horizon,  and  whose  given 
depth=rf;  then  A=EDxrf,  and  G=±d,  .:  AxG^EDx^which 
varies  as  ED  or  EC,  which  varies  as  AC.  Hence  the  strength 

AC 

is  as  -r-^,  that  is,  it  is  constant. 
AO 


W 


Fig.  127. 


If  the  sides  of  the  beam  be  parallel  planes,  and  its  longitudinal 
section  a  semi-parabola,  (see  Fig.  127,)  as  A'  C'  F'  D' ;  then  let 
d  equal  the  given  breadth,  and  we  have  A=c?xC'D',G=iC'D', 

C'D'1 

.'.  Ax G=i<ZxC'D'',  which  varies   as  C'D'1.     Hence  8x^7. 

But  since  A'D'  is  a  parabola,  A'C',  x  C'D", .-.  S  x 


MECHANICS.  161 

206.  Suppose  the  beams   to  be  cylinders  or  square  prisms, 
(resting  either  on  one  end  or  on  both  ends,)  whose  diameters  or 

D'  d' 

sides  are  D,  d,  then  S  :  s  :  :   -  —  —  —  •=.  :  Let  w'=0, 

LxfW+W    ZxiuH-io7 

D'  d* 

then  S:*::=:-,  which    expresses    the    relative 


strengths  of  two  cylindrical  beams  whose  lengths  are  L,  I,  diame- 
ters D,  d,  weights  W,  w,  the  former  of  which  supports  the  given 
weight  W  at  the  end  of  it,  and  the  latter  supports  only  its  own 
weight.  Let  d=D  ; 

1  1  1  2L 

S:S 


WxZ 

for,  since  w  :  W  :  :  I  :  L,  .-.  w=  —  —  .     In  this  case,  therefore,  the 

L 

ratio  of  S  :  s  expresses  the  relative  strengths  of  two  cylindrical 
beams  of  the  same  diameter,  one  of  which  supports  the  given 
weight  W  at  its  end,  and  the  other  supports  only  its  own  weight  ; 
and  if  the  beam  whose  length  is  L  breaks  when  W  is  placed  at 
the  end  of  it,  the  beam  whose  length  is  I  will  break  by  its  own 

1  2L 

weight.      Hence,   let    S=*.   then 


«     L'(W+2W)  T  /W+2W'\*    i 

.  •.  I  =  —  i_  _I  --  i  /.  Z=L  I  —  ^  —  J  =length  of  the  beam  of 

the  same  diameter  that  would  break  by  its  own  weight. 

207.  Let  the  beams  be  similar  cylinders,  then  D"  :  d3  :  :  L*  :  Z*, 
,:^::,^:^.     Andwhen 
2L«    ;  Wx/  L(W+aW)       If 

•  *w+w  ~"airf  OI      ~^vv  —    If' 


therefore,  a  cylindrical  beam  whose  length  is  L  breaks  with  the 
given  weight  W  placed  at  the  end  of  it,  a  similar  cylindrical 

L(W+2W) 
beam  whose  length  is—  —  —  -  -  will  break  with  its  own  weight. 

208.  If  a  horizontal  beam  be  supported  at  both  ends,  the  stress 
produced  by  its  own  weight,  W,  is  measured  by^LxW,  (Art. 
196.) 

If  the  beam  be  supported  at  one  end  only,  the  stress  is  meas- 

*  The  weights  of  similar  cylinders  of  the  same  density  are  as  the  cubes  of  their 
diameters  or  lengths  ;  therefore,  to  :  W  :  :  P  :  Ls  .•  w=  j-j-. 

21 


162  NATURAL   PHILOSOPHY. 

ured  by  the  whole  weight  applied  at  the  center  of  gravity,  and 
consequently  the  stress^LxW. 

Therefore  a  beam  supported  at  both  ends  has  four  times  the 
strength  of  the  same  beam,  supported  only  at  one  end.  And  if 
a  certain  beam  resting  on  one  end  breaks  by  its  own  weight,  a 
beam  of  the  same  dimensions  twice  as  long  will  break  by  its 
own  weight  when  resting  on  two  supports,  the  former  having 
just  four  times  the  strength  it  would  have  if  twice  as  long. 

If,  however,  instead  of  the  weight  of  the  beam  itself,  this  is 
left  out  of  the  account,  and  a  weight  W  be  appended,  then  the 
stress  on  the  beam  when  supported  at  one  end  will  be  measured 
by  L  x  W ;  while,  in  the  case  of  the  beam  supported  at  both 
ends,  (since  the  weight  being  at  the  center  is  also  at  the  center 
of  gravity  of  the  beam,)  the  stress  is  measured  as  before,  by  ^L 
xW'.  (Art.  200.)  Therefore,  a  weight  appended  at  the  end  of 
a  beam  supported  only  at  one  end  produces  four  times  the  stress, 
as  the  same  weight  applied  at  the  center  of  the  beam  when  sup- 
ported at  both  ends. 


209.  EXAMPLES. 

1.  What  must  be  the  length  of  a  beam  of  oak  one  inch  square, 
supported  at  both  ends,  which  is  just  capable  of  bearing  its  own 
weight  ? 

By  Art.  201,  a  beam  of  oak  1  foot  long  and  1  inch  square, 
weighing  £  pound,  just  supports  600  pounds.  And  by  Art.  206, 

(W+2W'\  * 
—  )  denotes  that  when  a  beam  whose 

length  is  L  breaks  when  W  is  placed  at  the  end  of  it,  I  is  the 
length  of  a  beam  that  will  break  with  its  own  weight ;  conse- 
quently, since  here  L=l,  W=£,  and  W-600,  Z=(i±i?2?)  = 

(2401)^=49  feet. 

2.  What  must  be  the  length  of  a  bar  of  iron  1  inch  square, 
supported  at  one  end,  which  would  break  by  its  own  weight  ? 

Here  L=l  foot,  W=3  pounds,  and  since  (Art.  208)  a  beam 
supported  at  one  end  will  break  with  \  as  great  a  weight  as 
when  supported  at  both  ends,  W'=547|  pounds, 


3.  Since  a  bar  of  iron  1  inch  square,  and  1  foot  long,  will  sup- 
port a  weight  of  2190  pounds,  what  must  be  the  dimensions  and 


MECHANICS 


163 


weight  of  a  similar  bar,  which  will  break  with  its  own  weight 
when  supported  at  both  ends  ? 

The  required  beam  being  similar  to  the  given  beam,  therefore, 
by  Art.  207,  its  length  equals  1461  feet.     And  L  (1)  :  Z(1461)  : : 
D  (1  inch) :  d=12 If  feet=the  side  of  its  square.    Again,  since  the 
weights  are  as  the  cubes  of  the  homologous  sides, 

w  :  W(3) : :  Z3(1461)3  :!,.•.  the  weight=9355605543  pounds. 

4.  Two  beams  are  of  equal  length  and  weight,  the  first  being 
a  square  prism  whose  section  is  4  inches  square,  the  second  an 
oblong  8  by  2  inches  :  How  much  stronger  is  the  second  beam 
than  the  first,  and  how  much  stronger  when  laid  on  the  narrow 
than  on  the  broad  side  ? 

Ans.  The  second  beam  is  TWICE  as  strong  as  the  first,  and  FOUR 
times  as  strong  when  laid  on  the  narrow,  as  on  the  broad  side. 

210.  On  the  foregoing  principles  Dr.  Gregory  makes  the  fol- 
lowing remarks,  most  of  which  were  originally  suggested  by  Gal- 
ileo, to  whom  we  are  indebted  for  the  earliest  investigation  of 
these  propositions.  From  the  preceding  deductions  (says  Greg- 
ory) it  follows,  that  greater  beams  and  bars  must  be  in  greater 
danger  of  breaking  than  less  similar  ones ;  and  that,  though  a 
less  beam  may  be  firm  and  secure,  yet  a  greater  similar  one  may 
be  made  so  long  as  necessarily  to  break  by  its  own  weight. 
Hence  Galileo  justly  concludes,  that  what  appears  very  firm,  and 
succeeds  well,  in  models,  may  be  very  weak  and  unstable,  or 
even  fall  to  pieces  by  its  weight,  when  it  comes  to  be  executed 
in  large  dimensions,  according  to  the  model.  From  the  same 
principles  he  argues  that  there  are  necessarily  limits  in  the  works 
of  nature  and  art,  which  they  cannot  surpass  in  magnitude  ;  that 
immensely  great  ships,  palaces,  temples,  &c.,  cannot  be  erected, 
since  their  yards,  beams,  bolts,  and  other  parts  of  their  frame, 
would  fall  asunder  by  their  own  weight.  Were  trees  of  a  very 
enormous  magnitude,  their  branches  would,  in  like  manner,  fall 
off.  Large  animals  have  not  strength  in  proportion  to  their  size ; 
and  if  there  were  any  land  animals  much  larger  than  those  we 
know,  they  could  hardly  move,  and  would  be  perpetually  sub- 
jected to  the  most  dangerous  accidents.  As  to  marine  animals, 
indeed,  the  case  is  different,  as  the  specific  gravity  of  the  water 
sustains  those  animals  in  a  great  measure  ;  and  in  fact  these  are 
known  to  be  sometimes  vastly  larger  than  the  greatest  land  ani- 
mals.* It  is  (says  Galileo)  impossible  for  Nature  to  give  bones 
to  men,  horses,  or  other  animals,  so  formed  as  to  subsist,  and  suc- 
cessfully to  perform  their  offices,  when  such  animals  should  be 
enlarged  to  immense  heights,  unless  she  uses  matter  much  firmer 

*  Whales  in  the  Northern  Regions,  are  sometimes  found  sixty  feet  long,  and 
weighing  seventy  tons. 


164  NATURAL   PHILOSOPHY. 

and  more  resisting  than  she  commonly  does ;  or  should  make 
bones  of  sfthickness  out  of  all  proportion  ;  whence  the  appearance 
and  figure  of  the  animal  must  be  monstrous.  Hence  we  natu- 
rally join  the  idea  of  greater  strength  and  force  with  the  grosser 
proportions,  and  that  of  agility  with  the  more  delicate  ones.  The 
same  admirable  philosopher  likewise  remarks,  in  connection  wit]? 
this  subject,  that  a  greater  column  is  in  much  more  danger  of  be- 
ing broken  by  a  fall,  than  a  similar  small  one ;  that  a  man  is  in 
greater  danger  from  accidents  than  a  child  ;  that  an  insect  can 
sustain  a  weight  many  times  greater  than  itself,  whereas  a  much 
larger  animal,  as  a  horse,  could  scarcely  carry  another  horse  of 
his  own  size.* 

211.  The  lateral  strengths  of  two  cylinders,  of  the  same  matter, 
and  of  equal  weight  and  length,  one  of  which  is  hollow,  and  the  other 
solid,  are  to  each  other  as  the  diameters  of  their  sections. 
'    Fig.  128. 


b 

—*um 
B 

Let  ABC,  abc,  (Fig.  128,)  ^represent  sections  of  two  cylinders, 
of  equal  length,  and  containing  equal  quantities  of  matter,  of 
which  ABC  is  hollow,  and  abc  is  solid.  Then  the  area  of  the  ring 
whose  breadth  is  AD,  is  equal  to  that  of  the  circle  abc.  But  ' 
the  strengths  of  these  areas  are  as  the  areas  .multiplied  by  the 
distances  of  their  centers  of  gravity  from  the  points  of  pressure 
A,  a*  (Art.  194  ;)  or,  since  the  areas  are  equal,  the  strengths  are 
as  AG  :  ag,  that  is,  as  the  diameters  of  their  sections. 

The  strongest  form,  therefore,  in  which  a  given  quantity  of 
matter  can  be  disposed,  is  that  of  a  hollow  cylinder ;  and  leaving 
out  of  view  the  diminished  rigidity  of  their  structures  or  fabrics, 
there  would  seem  to  be  no  limits  to  the  strength  which  might  be 
given  to  such  a  cylinder  by  increasing  its  diameter.  But  the 
proposition  is  true  only  when  the  sections  are  perfectly  circular  ; 
and  this  condition,  connected  with  the  want  of  rigidity  when  the 
annulus  becomes  very  thin,  occasions  limits  to  the  actual  opera- 
tion of  the  principle. 

212.  From  this  proposition  Galileo  justly  concludes,  that  Na- 
ture in  a  thousand  operations  greatly  augments  the  strength  of 

*  Gregory's  Mechanics,  I,  110. 


MECHANICS.  165 

substances  without  increasing  their  weight  ;  as  is  manifested  in 
the  bones  of  animals,  and  the  feathers  of  birds,  as  well  as  in  most 
tubes  or  hollow  trunks,  which,  though  light,  greatly  resist  any 
effort  to  bend  them.  Thus,  (says  he,)  if  a  wheat  straw,  which 
supports  an  ear  heavier  than  the  whole  stalk,  were  made  of  the 
same  quantity  of  matter,  but  solid,  it  would  bend  or  break  with 
far  greater  ease  than  it  now  does.  And  with  the  same  reason, 
art  has  observed,  and  experience  confirmed  the  fact,  that  a  hol- 
low cane,  or  tube  of  wood  or  metal,  is  much  stronger  or  firmer, 
than  if,  while  it  continues  of  the  same  weight  and  length,  it  were 
solid  ;  as  it  would  then,  of  consequence,  be  not  so  thick.  For 
the  same  reason,  lances,  when  they  are  required  to  be  both  light 
and  strong,  are  made  hollow.* 

213.  The  area  ABC  :  abc  :  :  AB2  :  ab2. 
And  DEF  :  abc  :  :  DE2  :  ab2. 
.-.  ABC-DEF  :  abc  :  :  AB2  -DE2  :  ab2. 

Or  the  area  of  the  ring  is  to  the  area  of  the  solid  section,  as 
AB*-DE2  :  ab*.  If  the  area  of  the  ring  is  equal  to  the  solid 
section,  then  AEa-DW=ab\  and  a6=N/(AB!!-DE  ). 

What  weight  could  be  sustained  at  the  middle  point  of  a  cylin- 
drical iron  tube  8  feet  long,  whose  diameter  is  H  inches,  and 
thickness  i  of  an  inch  ;  supposing  the  tube  to  be  supported  at 
both  ends  ? 

The  diameter  of  a  solid  cylinder  of  the  same  length  and 
weight=l.ll  inches,  and  (Art.  200) 

S  •    +  •  •  AxG     ax  -532      .3927 

:: 


Ixw'  ''  8x        '   1720' 

Therefore  W=291.3  lbs.=weight  which  would  be  sustained  by 
a  solid  cylinder  containing  the  same  quantity  of  matter  as  the 
tube,  and  which  consequently  measures  the  strength  of  the 
cylinder.  But,  (Art.  211,)  putting  S  for  the  strength  of  the  tube 
and  S'  for  that  of  the  cylinder, 

S  :  S'  :  :  |  :  1.11,  .-.  S  :  291.3  :  :  f  :  1.11,  .-.  8=393.64  Ibs. 

*  Gregory,  I,  112. 

t  Since  a  bar  of  iron  1  foot  long  and  1  inch  square,  weighs  2190  Ibs.,  (Art.  201,)  a 
cylinder  of  the  same  dimensions  weighs  1720  Ibs. 


166  NATURAL  PHILOSOPHY. 


PART   II. PRACTICAL    APPLICATIONS    OF    THE    PRINCIPLES    OP 

MECHANICS. 


v  CHAPTER  I. 

OF  THE  MECHANICAL  PROPERTIES  OF  MATTER. 

214.  Matter  constitutes  the  great  subject  of  Chemistry ;  mo- 
tion, that  of  Mechanical  Philosophy.     Chemistry  inquires,  first, 
whether  a  given  body  is  simple  or  compound, — whether  it  con- 
sists of  one  kind  of  matter,  or  of  two  or  more  different  kinds  of 
matter  united  in  one  body  ;  and,  secondly,  what  are  the  peculiar 
properties  of  each  individual  body.     Mechanical  Philosophy,  on 
the  other  hand,  takes  cognizance  of  those  properties  of  matter 
only  which  belong  either  to  all  bodies  whatsoever,  or  at  least  to 
extensive  classes  of  bodies.     The  changes  it  contemplates  are 
those  which  appertain  to  the  quantity,  position,  figure,  or  motion 
of  bodies,  while  it  leaves  to  Chemistry  all  those*  changes  which 
alter  the  constitution  of  bodies,  transforming  them  into  some- 
thing of  a  different  nature  from  what  they  were  before. 

215.  The  leading  mechanical  properties  of  matter  are  Divisi- 
bility, Porosity,  Compressibility,  Elasticity,  Indestructibility,  and 
Attraction. 

DIVISIBILITY. — Matter  is  susceptible  of  mechanical  division 
beyond  any  known  limits.  It  is  said  that  a  grain  of  musk  is 
capable  of  perfuming  for  several  years  a  chamber  twelve  feet 
square,  without  sustaining  any  sensible  diminution  of  its  volume 
or  weight.  Such  a  chamber  contains  nearly  3,000,000  cubic 
inches,  and  as  the  odor  of  the  musk  pervades  every  part  of  the 
room,  a  certain  number  of  particles  are  contained  in  each  cubic 
inch.  The  air  of  the  room  may  be,  in  the  mean  time,  changed 
many  thousand  times,  and  a  new  supply  of  odorous  particles 
furnished  to  each  successive  portion  of  air.  Hence  the  number 
of  particles  diffused  in  the  time  supposed  exceeds  all  computa- 
tion, and  yet  the  weight  of  the  material  is  not  sensibly  dimin- 
ished. The  thickness  of  a  soap-bubble,  according  to  Newton,  the 
moment  before  it  bursts,  is  only  thp  four-millionth  part  of  an 
inch.  The  thread  of  a  silk-worm  is  a  perfectly  smooth  cylinder 
whose  diameter  is  nearly  the  two  thousandth  part  of  an  inch, 
and  yet  tne  spider's  web  is  vastly  more  attenuated. 


MECHANICS.  167 

216.  POROSITY. — In  many  bodies  the  pores,  or  vacant  spaces, 
are  easily  distinguishable  by  the  naked  eye,  as  in  the  case  of 
sponge,  wood,  and   most  kinds   of  stones.     Many  substances 
which  do  not  exhibit  pores  to  the  naked  eye,  still  betray  them  to 
the   microscope.     Metals   do   not  usually,  when   pure,  appear 
porous,  even  under  the  microscope,  but  still  such  a  structure  may 
be  detected  by  mechanical  means.     Thus  if  a  hollow  ball  of 
gold  be  filled  with  water,  plugged  close,  and  compressed  in  a 
vise,  the  water  will  exude  through  the  metal.     By  means  of  this 
structure,  in  animals  and  vegetables,  air  and  water  circulate 
freely  through  them,  aiding  the  functions  of  life,  as  the  sap  in 
trees.     A  cross  section,  or  thin  slice  of  wood,  viewed  with  the 
microscope,  shows  that  the  pores  occupy  usually  a  much  greater 
space  than  the  solid  matter  of  the  wood.     Indeed,  the  solid  par- 
titions between  the  larger  cells  are  themselves  seen,  by  powerful 
magnifiers,  to  be  full  of  pores.     The  surface  of  the  body  of  a 
middle-sized    man  has   been  estimated  to  contain  more  than 
20,000,000  pores,*  the  skin  being  perforated  with  a  thousand 
holes  to  every  inch.     Wood  consists  of  bundles  of  fibres  of  dif- 
ferent degrees  of  fineness,  usually  aggregated  together  so  loosely,  *' 
that  a  free  circulation  of  water  is  easily  maintained  between 
them.     Glass  is  the  only  solid  known  which  appears,  as  far  as 
experiments  have  gone,  to  be  absolutely  impermeable   to  all 
fluids.f 

217.  COMPRESSIBILITY. — All  bodies  yield  more  or  less  to  external 
pressure,  undergoing  a  diminution  of  volume  proportional,  in 
each  case,  to  the  force  applied.  Aeriform  bodies,  as  common 
air,  yield  readily  to  any  compressing  force,  the  diminution  of 
volume  being  always  exactly  proportional  to  that  force.     Sol- 
ids, also,  as  wood  and  stone,  are  compressed,  in  different  de- 
grees, under  heavy  weights.     A  cork  immersed  two  hundred  feet 
in  the  sea,  is  so  much  compressed  as  to  become  heavier  than 
water  and  to  sink ;  and  a  pint  bottle  of  fresh  water,  corked  closely, 
and  sunk  to  a  great  depth  in  the  ocean,  will,  when  drawn  up,  be 
found  to  be  filled  with  salt  water.     This  remarkable  fact  is  ex- 
plained by  supposing  that  the  cork  has  been  so  much  contracted 
in  bulk  as  to  admit  the  salt  water,  which  being  heavier  than  the 
fresh,  displaced  it  and  occupied  the  bottle.    As  the  cork  sustained 
an  equal  pressure  on  all  sides,  it  would  not  be  removed  out  of 
its  place  ;  and,  as  the  bottle  was  drawn  up,  and  the  pressure  was 
diminished,  the  cork  would  regain  its  original  dimensions.    Hard 
mineral 'substances,  as  blocks  of  granite,  indicate  some  contrac- 
tion of  volume  when  subjected  to  the  pressure  of  high  and 
massive  walls.     Liquids  resist  compression  much  more  than 
either  air  or  solid  bodies.     Still,  under  enormous  weights,  it  may 
be  rendered  sensible,  as  will  be  more  fully  explained  hereafter. 

*  Leslie,  Nat.  PhiL  I,  18.  t  Pouillet,  EL  Phys.  1. 1,  29. 


168  NATURAL   PHILOSOPHY. 

218.  ELASTICITY. — Bodies  are  said  to  be  perfectly  elastic  when 
IJhey  restore  themselves  to  their  original  dimensions  when  re* 
leased,  and  with  a  force  equal  to  that  with  which  they  were 
compressed.     Air  and  all    gases  are  of  this  class ;  and  even 
liquids,  as  water,  are  found  to  conform  to  the  same  law,  and,  in 
this  sense  therefore,  they  must  also  be  regarded  as  perfectly 
elastic  substances.*     Metals,  indeed,  have  the  same  property,  a 
double  extension  or  compression,  requiring  twice  the  force ;  triple, 
three  times  the  force,  and  so  on.     The  elasticity  of  wood  is  ex- 
emplified in  the  cross-bow,  and  that  of  a  mineral  substance  pecu- 
liarly in  mica.     The  elasticity  of  torsion,  or  the  force  by  which  a 
wire  when  twisted  endeavors  to  resume  its  natural  state,  is  em- 
ployed as  the  most  delicate  test  and  measure  of  force  known, 
the  force  of  torsion  being  always  proportional  to  the  angle 
through  which  the  body  has  been  twisted. 

219.  INDESTRUCTIBILITY. — Matter  is  wholly  indestructible.     In  all 
the  changes  we  see  going  on  in  bodies  around  us,  not  a  particle 
of  matter  is  lost ;  it  merely  changes  its  form ;  nor  is  there  any 
reason  to  believe  that  there  is  now  a  particle  of  matter  either 
more  or  less  than  there  was  at  the  creation  of  the  world.    When 
we  boil  water  and  it  passes  into  the  invisible  state  of  steam, 
this,  on  cooling,  returns  again  to  the  state  of  water  without  the 
least  loss  ;  when  we  burn  wood,  the  solid  matter  of  which  it  is 
composed,  passes  into  different  forms,  some  into  smoke,  some  into 
different  kinds  of  airs,  or  gases,  some  into  steam,  and  some  re- 
mains behind  in  the  state  of  ashes.     If  we  should  collect  all  these 
various  products,  and  weigh  them,  we  should  find  the  amount  of 
their  united  weights  the  same  as  that  of  the  body  from  which 
they  were  produced,  so  that  no  portion  is  lost.     Each  of  the  sub- 
stances into  which  the  wood  was  resolved,  is  employed  in  the 
economy  of  nature  to  construct  other  bodies,  and  may  finally 
re-appear  in  its  original  form.     In  the  same  manner,  the  bodies 
of  animals,  when  they  die,  decay  and  seem  to  perish  ;  but  the 
matter  of  which   they  are  composed  merely  passes  into  new 
forms  of  existence,  and  re-appears  in  the  structure  of  vegetables 
or  other  animals. 

220.  ATTRACTION. — This  property  of  matter  produces  or  governs 
a  large  part  of  the  phenomena  of  the  natural  world.     By  it  all 
matter  tends  toward  all  other  matter,  and  by  it  the  particles  of 
matter  unite,  forming  innumerable  compounds.     It  is  chiefly  this 
property  in  different  degrees  which  constitutes  the  STRENGTH  OF 
MATERIALS, — a  subject  which  has  been  already  considered  theoreti- 
cally, (Arts.  193,  213,)  but  which  it  will  be  useful  now  to  con- 
sider practically,  in  its  relation  to  the  arts. 

*  Mosely's  Illustrations,  Sec.  32. 


MECHANICS.  169 

221.  The  strength  of  substances  in  the  direction  of  the  length 
has  been  determined,  experimentally,  by  suspending  small  cylin- 
ders or  wires  of  each  material  vertically,  and  applying  weights 
at  the  bottom  until  they  broke.     Of  all  substances,  that  which 
sustains  the  greatest  load  is  iron.     Different  materials,  before 
rupture,  increase  in  length  in  differenUdegrees.     Bars  of  the  best 
wrought  iron  are  elongated  about  .000082  for  a  load  of  one  ton 
to  the  square  inch.     Iron  in  the  form  of  wires  admits  of  a  greater 
extension.     A  bundle,  or  cable  of  small  wires,  will,  under  similar 
circumstances,  be  elongated  .000091  ;  and  becomes  more  exten- 
sible in  proportion  as  the  wires  are  smaller.     Bar  iron  will  bear 
to  be  extended  .000714  without  injury ;  and  several  kinds  of 
wood,  as  oak,  pine,  and  fir,  will  bear  an  elongation  three  times 
as  great.     Iron  wire,  on  account  of  its  extraordinary  tenacity, 
has  within  a  few  years  been  most  successfully  applied  to  the 
construction  of  suspension  bridges.   Iron  wire  ~  inch  in  diameter, 
has  been  found  by  experiment  capable  of  bearing  a  load  of  60 
tons  to  the  square  inch  without  breaking.    In  one  instance,  indeed, 
the  load'  sustained  was  90  tons.     The  Menai  bridge,  in  Wales, 
one  of  the  most  celebrated  works  of  the  age,  is  supported  by  iron 
wire  cables.     Its-  span  between  the  points  of  suspension  is  560 
feet,  its  height  above  highwater  mark  100  feet,  and  the  roadway 
30  feet  in  breadth.     Its  weight  is  over  four  millions  of  pounds, 
(2000  tons,)  but  the  whole  is  suspended  by  four  lines  of  strong  iron 
cables  by  perpendicular  rods  5  feet  apart.     Russia  bar  iron  has 
a  tenacity  of  27  tons  to  the  square  inch.     The  best  cast  iron  is 
about  one  third  as  strong,  having  a  tenacity  of  about  9  tons  to 
the  inch.     The  tenacity  of  platina  wire  is  nearly  as  great  as  that 
of  bar  iron.     The  comparative  strength  of  several  substances 
much  used  in  the  arts,  is  thus  stated  by  Mosely :  7  rods  of  ma- 
hogany, taken  together ;  5  of  pine,  oak,  or  beach ;  3  of  box  or 
of  cast  iron  ;  2  of  gold  ;  li  of  silver  or  copper,  have  respectively 
the  same  tenacity  as  1  corresponding  rod  of  wrought  iron  ;  or  as 
a  rod  whose  section  is  T\  made  of  steel  or  fine  wire  cable. 

222.  As  the  materials  used  in  building  are  liable  to  give  way 
by  the  superincumbent  weight,  it  is  often  important  to  know  the 
relative  power  of  different  materials  to  resist  forces  tending  to 
crush  them.     Numerous  experiments  on  this  subject  have  been 
instituted  by  taking  small  blocks  of  similar  shape  and  size,  and 
loading  them  respectively  with  weights  until  they  were  crushed. 
It  appears  that  cast  iron  is  best  adapted  of  all  the  materials  in 
common  use  to  sustain  such  pressures  ;  that  bar  iron  is  not  more 
than  half  as  strong ;  granite,  one  sixth ;  Italian   marble,  one 
seventh  ;  free-stone,  one  tenth  ;  brick  work,  still  less.     Wooden 
columns  have  comparatively  little  power  of  resisting  a  force 
tending  to  crush  them  in  the  direction  of  their  fibres.     Short 
columns,  however,  bear  stronger  pressures  than  long  ones,  the 

22 


170  NATURAL   PHILOSOPHY. 

strength  diminishing  in  a  geometrical,  while  the  height  is  in- 
creased in  an  arithmetical  ratio.  Slight  changes  of  form  are 
found  sometimes  greatly  to  affect  the  strength  of  a  column. 
Thus  merely  rounding  the  ends  of  a  perpendicular  column  makes 
its  strength  only  one  third  that  of  a  column  whose  extremities 
are  flat.  If  we  take  thr^b  columns,  equal  in  all  respects,  and 
round  both  ends  of  the  first,  one  end  of  the  second,  and  leave 
both  ends  of  the  third  flat,  their  respective  strengths  will  be  as 
the  numbers  1,  2, 3.  The  shape  of  a  column  has  great  influence 
on  its  strength.  A  cylindrical  column  is  weakest  in  the  middle  ; 
and  it  is  found  that  the  strength  of  a  column  of  cast  iron,  con- 
taining a  given  weight  of  metal,  whether  it  be  solid  or  hollow, 
is  much  greater  when  it  is  cast  in  the  form  of  a  double  cone, 
that  is,  with  its  greatest  thickness  in  the  middle  of  its  height, 
and  tapering  to  its  extremities,  than  when  cast  in  any  other  form. 
In  lofty  stone  columns,  however,  their  own  weight  may  consti- 
tute a  large  part  of  the  load  to  be  sustained,  and  hence  it  is  the 
practice  of  architects  to  make  the  swell  below  the  center.  In 
some  of  the  Grecian  temples,  it  was  one  third  from  the  base  ; 
and  this  rule  is  now  frequently  adopted. 

223.  In  practice,  materials  cannot  safely  be  subjected  to  con- 
stant strains  or  thrusts  approaching  to  those  which  produce  rup- 
ture. They  are  liable  to  various  occasional  and  accidental 
pressures  ;  and  others  of  a  permanent  kind,  resulting  from  the 
settling  of  the  pile.  It  is  therefore  regarded  as  not  entirely  safe 
to  load  any  structure  of  stone  more  than  one  sixth  the  amount 
of  the  pressure  that  crushes  it.  Iron,  cast  or  wrought,  may  be 
loaded  to  one  fourth  that  amount. 

The  view  taken  by  Barlow  of  the  mode  in  which  a  horizontal 
beam  undergoes  compression,  leads  to  results  somewhat  different 
from  those  investigated  in  Articles  193,  &c.  When  a  beam  is  bent 
in  the  middle,  the  fibres  on  the  upper  side  undergo  compression, 
while  those  on  the  under  side  undergo  extension,  as  in  Fig.  129, 
Fig.  129. 


and  between  the  two  is  a  line  that  sustains  neither,  which  is 
called  the  neutral  axis.  -Since,  throughout  its  neutral  axis,  the 
strength  of  the  beam  is  not  at  all  called  into  action,  this  will  not 
be  impaired  by  boring  a  hole  through  the  beam  in  the  direction 
of  that  axis.  What  constitutes  the  strength  of  a  beam  is  its  re- 
sistance to  extension  on  the  lower,  and  to  compression  on  the 
upper  side.  These  act  as  antagonist  forces,  and  if  either  of 
them  yields,  the  beam  is  broken.  Hence  the  strength  of  the 


MECHANICS.  171 

beam  is  not  impaired  by  sawing  it  through  the  upper  side  as  low 
as  to  the  neutral  axis.  This  extends  to  about  five  eighths  of  the 
depth.* 

224.  Although  wood  has  not  intrinsically  the  strength  of  iron 
or  stone,  yet  its  lightness  in  some  measure  compensates  for  this, 
so  that  large  structures  of  wood  have  sometimes  great  power  of 
resistance  to  external  forces.  Pine  is  only  one  fifteenth  as 
heavy  as  cast  iron,  while  it  has  more  than  half  the  tenacity. 
Sixteen  bars  of  it  would  weigh  only  the  same  as  one  bar  of 
wrought  iron,  while  they  would  have  three  times  the  strength. 
Many  large  structures,  when  constructed  of  heavy  materials,  are 
weak  from  their  own  weight.  Iron  roofs  have  been  known  to 
fall  in  by  their  own  pressure.  Trees  often  resist  the  action  of 
external  forces  which  overturn  works  of  art  apparently  of  much 
more  stable  materials  ;  and  nothing  is  more  deserving  the  atten- 
tion of  the  architect  than  the  rules  which  nature  has  observed, 
both  in  the  selection  and  distribution  of  the  materials  of  which 
trees  are  constructed.  The  tapering  form  of  their  trunks  ;  the 
increased  diameter  and  density  of  their  bases ;  the  buttresses 
that  frequently,  in  a  large  tree  especially,  support  the  trunk  on 
every  side  ;  the  comparative  lightness  of  the  extended  top ;  and 
the  universal  symmetry  of  form  that  pervades  the  entire  struc- 
ture ;  these  qualities,  severally  and  collectively,  add  to  the 
strength  of  trees,  and  fit  them  to  encounter  the  most  violent 
winds.  Indeed,  mechanical  writers,  when  they  have  descended 
to  a  minute  investigation  of  the  structure  of  trees,  have  found 
the  most  refined  use  made  of  such  mechanical  principles  as  tend 
both  to  the  greatest  strength  and  greatest  economy  of  material. 
Thus,  in  a  cylinder  of  wood,  like  the  trunk  of  a  tree,  the  neutral 
axis  is  near  the  center  ;  hence  the  resistance  to  compression  on 
one  side  and  to  that  extension  on  the  other,  act  in  opposition  to 
each  other,  when  a  tree  is  bent  by  the  wind,  and  thus  the  trunk  is 
prevented  from  breaking. 


CHAPTER  II. 

GENERAL  OBSERVATIONS  ON  MOTION. 

225.  MOTION  and  rest  are  accidental  states  of  bodies,  nor  is  a 
body  naturally  prone  to  one  state  more  than  to  the  other.  If  it 
is  found  at  rest,  it  is  because  it  is  kept  in  equilibrium  by  opposite 

*  Mosely. 


172  NATCRAL   PHILOSOPHY. 

and  equal  forces  ;  and  if  it  is  found  in  motion,  it  is  because  it 
has  been  put  in  motion  by  some  force  'extrinsic  to  itself.  The 
resistances  to  motion  which  exist  near  the  surface  of  the  earth, 
particularly  gravity,  create  a  seeming  tendency  to  a  state  of 
rest  ;  but,  in  reality,  rest  is  no  more  the  natural  state  of  bodies 
than  motion  is. 

226.  Motion  is  distinguished  into  absolute  and  relative.     Abso- 
lute motion,  is  a  change  of  place  in  space  with  respect  to  any 
fixed  point :  Relative  motion,  is  a  change  of  place  in  bodies  with 
respect  to  each  other.     A  body  may  be  at  the  same  time  in  a 
state  of  absolute  motion,  and  of  relative  rest.     Thus,  all  the  dif- 
ferent articles  contained  in  a  ship  under  sail,  have  a  motion  in 
common  with  the  ship,  but  may  be  at  rest  with  respect  to  each 
other.     When  a  man  walks  toward  the  stern  of  a  ship  at  the 
same  rate  as  that  of  the  ship,  he  is  in  motion  with  respect  to  the 
ship,  but  at  rest  with   respect  to  the  earth.     When  a  balloon, 
carried  along  by  the  wind,  attains  the  same  velocity  as  the  wind, 
it  is  relatively  at  rest,  and  appears  to  the  aeronaut  to  be  in  a 
perfect  calm,  although  it  may  be  actually  moving  one  hundred 
miles  an  hour.     Since  the  earth  in  its  annual  revolution  around 
the  sun,  is  moving  eastward  at  the  rate  of  nineteen  miles,  or 
100,000  feet  per  second,  were  a  cannon  ball,  at  a  certain  time 
of  day,  fired  eastward  at  the  rate  of  2000  feet  per  second,  the 
only  effect  would  be  to  add  2000  feet  to  the  velocity  which  the 
ball  had  before  in  common  with  the  earth :  and  were  it  fired 
westward,  the  effect  would  be  merely  to  stop  2000  out  of  100,000 
parts  of  its  previous  motion,  while  the  cannon  would  proceed 
onward,  leaving  it  behind.*     Did  not  the  atmosphere  partake 
of  the  diurnal  motion  of  the  earth,  but  were  it  to  remain  at  rest 
with  respect  to  this  motion,  the  progress  of  any  place  to  the  east, 
would  cause  a  relative  motion  of  the  air,  or  a  wind,  westward, 
which  would  blow  with  a  violence  far  surpassing  that  of  the 
most  terrible  hurricanes,  f 

Indeed,  we  cannot  be  sure  that  we  have  ever  seen  a  body  ab- 
solutely at  rest.  In  our  stillest  moments,  we  are  revolving  with 
the  earth  on  its  axis ;  we  are  accompanying  the  earth  in  its 
annual  revolution  from  west  to  east  around  the  sun ;  and  are 
perhaps  attending  the  solar  system  around  a  common  center  of 
motion.J 

227.  Apparent  motion,  as  distinguished  from  relative,  is  that  in 
which  the  body  that  seems  to  be  moving  is  quiescent,  and  the 
motion  is  owing  to  a  real  motion  in  the  spectator.     Thus,  the 

*  Robinson's  Mechanical  Phil.  I,  31. 

t  Winds  are  in  fact  frequently  produced  by  this  cause,  viz.  by  their  having  a  rela- 
tive velocity  different  from  that  of  the  part  of  the  earth  over  which  they  blow, 
t  Young's  Natural  Phil.  1, 19. 


MECHANICS.  173 

backward  motion  of  the  trees  to  one  riding  rapidly,  the  receding 
of  the  shore  to  one  who  is  sailing  from  it  with  a  fair  wind,  and 
the  diurnal  motions  of  the  heavenly  bodies  from  east  to  west,  in 
consequence  of  the  revolution  of  the  spectator  in  the  opposite 
direction  ;  these  are  severally  examples  of  apparent  motion.  It 
is  often  a  very  difficult  problem  to  deduce  the  real  from  the  ap- 
parent motion.  While  a  planet,  as  Venus,  is  revblving  about 
the  sun  in  an  orbit  nearly  circular,  its  motions,  as  seen  from  the 
earth,  are  extremely  irregular ;  and  to  make  all  these  irregu- 
larities consistent  with  the  real  motion,  has  been  a  perplexing 
problem  in  astronomy.  We  can  sometimes  decide  that  a  given 
motion  is  real,  because  we  observe  a  cause  in  operation,  which 
is  competent  to  produce  it.  The  impulse  of  the  wind,  or  the  di- 
rection of  the  current,  will  satisfactorily  account  for  a  ship's  re- 
ceding from  a  given  object,  while  no  cause  appears  why  the 
object  should  recede  from  the  ship  ;  the  revolution  of  the  earth 
on  its  axis  is  a  cause  competent  to  explain  the  apparent  revolu- 
tion of  the  heavens,  while  we  can  find  no  cause  for  their  actual 
revolution.  The  effects  also  of  a  given  motion,  enable  us  to  de- 
cide whether  it  is  real  or  apparent.  Thus  a  constant  tendency 
to  move  in  a  straight  line  is  characteristic  of  real  motion.* 

228.  The  LAWS  OF  MOTION  have  been  already  recited  in  Chap- 
ter I,  and  concise  illustrations  of  them  were  added  in  that  place. 
It  was  necessary  to  proceed  thus  far  at  the  beginning  of  this 
work,  since  these  constitute  the  fundamental  principles  of  me- 
chanics.    By  their  great  comprehensiveness,  they  furnish  the 
most  convenient  classification  of  the  various  phenomena  of  mo- 
tion, and  it  will  therefore  be  useful  to  resume  the  consideration 
of  them.     They  are  very  remarkable  examples  of  a  happy  gen- 
eralization ;  but  their  very  comprehensiveness  renders  them  dif- 
ficult to  be  understood  by  the  young  learner ;  nor  can  they  be 
thoroughly  mastered  in  all  their  relations,  until  after  considera- 
ble proficiency  is  made  in  the  science  of  Mechanics.    These  laws 
indeed  are  the  chief  foundation  of  Newton's  Principia.f 

229.  FIRST  LAW. — A  body  continues  always  in  a  state  of  rest,  or 
of  uniform  rectilinear  motion,  until,  by  some  external  force,  it  is 
made  to  change  its  state. — This  law  contains  the  doctrine  of  INER- 
TIA, expressed  in  four  particulars.     First,  that  unless  put  in  mo- 
tion by  some  external  force,  a  body  always  remains  at  rest ; 
secondly,  that  when  once  in  motion,  it  continues  always  in  mo- 
tion, unless  stopped  by  some  force  ;  thirdly,  that  this  motion, 
arising  from  INERTIA,  is  uniform  ;  and,  fourthly,  that  this  motion 
is  in  right  lines.     The  proofs  by  which  this  and  the  other  laws 
of  motion  are  established,  have  been  already  stated.     (Art.  22.) 

*  Wood's  Mechanics,  p.  22.  r  Young's  Nat.  Phil.  I,  26. 


174  NATURAL   PHILOSOPHY. 

It  is  our  present  object  to  make  the  application  of  these  laws  to 
various  phenomena  of  nature  and  art. 

230.  And  first,  with  respect  to  bodies  at  rest.     The  operation 
of  this  principle  is  seen,  when  a  horse  starts  suddenly  forward, 
and  his  rider  is  thrown  backward.     "  When  we  desire  a  person, 
with  suspected  disease  of  the  brain,  to  shake  his  head,  and  tell 
whether  he  feels  pain,  we  are  doing  nearly  the  same  as  if  we 
touched  the  naked  brain  with  the  finger  to  find  the  tender  part, 
for  the  inertia  of  the  brain,  when  the  skull  is  moved,  causes  a 
momentary  pressure  between  it  and  the  skull,  almost  equivalent 
for  our  purpose  to  such  a  touch."*     In  consequence  of  the  inertia 
of  matter,  before  a  body  can  be  brought  to  the  required  velocity, 
this  velocity  must  be  impressed  upon  every  particle  of  matter  it 
contains.     Hence,  the  more  numerous  its  particles,  the  greater 
its  inertia,  which  is  therefore  proportioned  to  the  quantity  of 
matter.     But  the  weight  also  is  proportioned  to  the  quantity  of 
matter,  and  therefore  the  inertia  is  proportioned  to  the  weight. 
Yet  it  must  be  carefully  distinguished  from  weight,  having  in 
fact  nothing  in  common  with  it,  except  that  both  are  proportion- 
ed to  the  quantity  of  matter,  and  of  course  to  each  other.     But 
were  we  to  strike  with  a  hammer  upon  the  top  of  a  body  falling 
toward  the  earth,  the  resistance  from  inertia  would  be  the  same 
as  if  the  body  were  struck  with  the  same  force  on  the  side  ;  or 
in  whatever  direction  the  blow  were  applied,  a  similar  resistance 
would  be  felt.    This  seems  little  else  than  what  we  commonly  un- 
derstand by  the  reaction  of  a  body  ;  but  we  conceive  this  reaction 
itself  to  depend  upon  an  inherent  property  in  matter,  to  which 
we  give  the  name  of  Inertia.     Inertia  is  the  cause  and  reaction 
the  effect.     A  vast  weight  may  be  moved  on  a  horizontal  rail- 
way by  a  comparatively  small  force,  provided  it  can  be  got  in 
motion  with  the  required  velocity.     In  transporting  large  quan- 
tities (eighty  tons,  for  instance)  of  coal,  the  weight  is  distributed 
into  a  number  of  different  cars,  connected  together  by  a  loose 
chain,  in  order  that  the  inertia  of  the  several  parts  may  be  over- 
come successively.! 

231.  In  consequence  of  the  inertia  of  matter,  the  motion  ap- 
plied to  a  body  does  not  instantly  pervade  the  mass.     In  order 
to  this,  motion  must  be  applied  gradually,  especially  if  the  body 
is  large  ;  for  if  it  is  applied  suddenly,  it  is  frequently  all  ex- 
pended on  a  part  only  of  the  mass,  the  cohesion  is  overcome, 
and  the  body  is  broken.     This  explanation  accounts  for  several 
familiar  facts.    When  a  team  starts  suddenly  with  a  heavy  load, 
the  effort  is  either  wholly  ineffectual,  or  some  part  of  the  har- 

»  Arnott's  El.  Phys.  p.  50. 

t  See  account  of  the  Hatton  Railway  in  "  Strickland's  Reports." 


MECHANICS.  175 

ness  or  tackling  gives  way.  If  we  draw  a  heavy  weight  by  a 
slender  string,  a  slow  and  steady  pull  will  move  the  weight, 
when  a  sudden  twitch  would  break  the  string  without  starting 
the  mass.  The  same  principle  applies  to  bodies  already  in  mo- 
tion. Thus,  when  a  horse  in  a  carriage  starts  suddenly  forward, 
he  may  break  loose  as  well  when  the  carriage  was  previously 
in  motion,  as  when  it  was  at  rest.  The  inertia  of  a  body  is  in 
fact  the  same  whether  the  body  is  in  motion  or  at  rest,  opposing 
the  same  resistance  to  its  moving  with  increased  velocity,  as  to 
its  beginning  to  move  from  a  state  of  rest.  Several  singular 
phenomena  result  from  the  same  cause,  showing  that  time  is 
necessary  in  order  that  motion  communicated  by  impulse,  may 
pervade  an  entire  mass.  A  pistol  ball  fired  through  a  pane  of 
glass,  frequently  makes  a  smooth,  well-defined  hole,  and  does 
not  fracture  the  other  parts  of  the  glass.  Here,  the  momentum 
of  the  ball  is  communicated  to  the  particles  of  glass  immediately 
before  it.  Had  the  impulse  been  gradual,  the  same  motion 
would  have  diffused  itself  over  the  whole  pane,  and  every  part 
would  have  felt  the  shock.  A  ball  fired  through  a  board  deli- 
cately suspended,  causes  ne  vibrations  in  the  board.  A  cannon 
ball,  having  very  great  velocity,  passes  through  a  ship's  side, 
and  leaves  but  little  mark,  while  one  with  less  speed,  splinters 
and  breaks  the  wood  to  a  considerable  distance  around.  A  near 
shot  thus  often  injures  a  ship  less  than  one  from  a  greater  dis- 
tance.* A  soft  substance,  as  clay  or  tallow,  may  be  fired  through 
a  plank, — the  body,  by  its  great  momentum,  forcing  its  way 
through  the  plank,  before  the  motion  has  had  time  to  diffuse! 
itself  through  the  contiguous  parts.  The  whole  momentum  be- 
ing concentrated  upon  the  part  immediately  before  the  body,  the 
cohesion  of  that  part  is  destroyed. 

232.  Secondly,  let  us  consider  the  effects  of  inertia  as  it  re- 
spects bodies  in  motion.  All  bodies  in  contact  with  each  other 
acquire  a  common  motion ;  as,  for  example,  a  horse  and  his 
rider,  a  ferry-boat  and  its  passengers,  a  ship  and  every  thing 
within  it,  the  earth  and  all  things  on  its  surface.  Whenever 
either  of  these  bodies  stops  suddenly,  the  movable  bodies  con- 
nected with  it  are  thrown  forward.  Were  the  revolution  of  the 
earth  on  its  axis  to  be  suddenly  arrested,  the  most  dreadful  con- 
sequences would  ensue  ;  every  thing  movable  on  its  surface,  as 
water,  rocks,  cities,  and  animals,  not  receiving  instantaneously 
this  backward  impulse,  would  fly  off  eastward  in  promiscuous 
ruin.  Were  the  diurnal  motion  of  the  earth,  however,  very 
gradually  diminished,  until  it  finally  ceased,  so  that  time  should 
be  afforded  to  communicate  the  loss  by  slow  degrees  to  the 
bodies  on  its  surface,  no  such  effects  would  take  place.  If  a 
passenger  leaps  from  a  carriage  in  rapid  motion,  he  will  fall  in 

Arnott's  El.  Phys.  p.  104. 


176 


NATURAL   PHILOSOPHY. 


the  direction  in  which  the  carriage  is  moving  at  the  moment  his 
feet  meet  the  ground  ;  because  his  body,  on  quitting  the  vehicle, 
retains,  by  its  inertia,  the  motion  which  it  had  in  common  with 
it.  When  he  reaches  the  ground,  this  motion  is  destroyed  by 
the  resistance  of  the  ground  to  the  fget,  but  it  is  retained  in  the 
upper  and  heavier  part  of  the  body,  so  that  the  same  effect  is 
produced  as  though  the  feet  had  been  tripped. 

233.  Although,  on  account  of  the  numerous  impediments  to 
motion  which  exist  on  the  surface  of  the  earth,  bodies  are  una- 
ble to  maintain  for  any  considerable  time  the  motion  they  have 
acquired,  yet  we  see  the  first  law  of  motion,  so  far  as  it  respects 
the  tendency  of  bodies  to  persevere  in  motion,  fully  confirmed  in 
the  continued  and  unaltered  revolution  of  the  heavenly  bodies. 
These  are  impelled  by  no  renewed  forces,  but  revolve  from  age 
to  age  in  an  undeviating  course,  simply  because  they  meet  with 
no  impediments. 

X534.  Thirdly,  bodies,  in  consequence  of 
their  inertia,  have  a  tendency  to  move  over 
equal  spaces  in  equal  times,  that  is,  to  move 
uniformly.  In  a  ball  rolled  on  ice,  in  a  pen- 
dulum continuing  to  vibrate  after  the  mov- 
ing force  is  withdrawn,  and  in  numerous 
cases  similar  to  these,  we  observe  in  nature 
and  art  this  tendency  to  uniform  motion ; 
but  in  all  these  cases,  the  motion  is  not  ab- 
solutely uniform,  but  more  or  less  retarded 
by  the  resistances  encountered.  A  much 
nearer  approximation  to  the  truth  is  ob- 
tained by  means  of  a  piece  of  apparatus- 
called  Atwood's  Machine.  Its  construction, 
omitting  some  parts  not  essential  to  the 
principle,  is  as  follows.  The  triangular 
base  and  upright  pillars  (which  are  usually 
of  mahogany)  constitute  the  frame,  which  is 
surmounted  by  a  horizontal  table  or  plate  of 
wood  AB,  Fig.  130,  perforated  with  several 
holes.  C  is  a  vertical  wheel,  which,  by  a 
contrivance  called  friction  wheels,  (not  rep- 
resented in  the  figure,)  is  made  to  revolve 
with  the  least  possible  resistance  from  fric- 
tion. D  and  E  are  two  weights  exactly 
equal,  and  connected  by  a  slender  string 
passing  over  the  wheel  C.  FG  is  a  perpen- 
dicular scale  graduated  into  inches  from  top 
to  bottom,  extending  from  0  to  60  or  70, 
according  to  the  height  of  the  machine. 
H  is  a  movable  ring  which  slides  up  and 
down  on  the  scale,  and  K  is  a  brass  plate 


MECHANICS.  177 

sliding  in  the  same  manner.*  A  great  variety  of  the  prin- 
ciples of  motion  may  be  established  by  means  of  this  appa- 
ratus, but  we  are  at  present  concerned  only  with  the  method 
of  showing,  that  a  body  when  once  put  in  motion  continues, 
by  its  inertia,  to  move  uniformly,  after  the  moving  force  is  with- 
drawn. It  is  obvious  that  the  weights  D  and  E  balance  each 
other,  and  consequently,  that  the  power  of  gravity  is  entirely  re- 
moved from  D,  so  that  it  is  at  liberty  to  obey  the  full  and  exclu- 
sive influence  of  any  force  that  may  be  applied  to  it.  If  there- 
fore an  impulse  be  given,  by  the  finger,  for  example,  to  D,  when 
at  the  top  of  the  scale,  it  ought,  in  conformity  to  the  law  under 
consideration,  to  move  uniformly  along  down  the  scale,  passing 
over  the  same  number  of  inches  in  each  successive  second. 
Such  appears  to  be  the  fact.  But  in  order  to  give  still  greater 
precision  to  the  experiment,  a  small  brass  bar  is  laid  on  D,  which 
communicates  motion  to  it,  accelerating  its  progress  until  it 
comes  to  the  brass  ring  H.  where  the  bar  lodges,  and  the  weight 
D,  after  it  leaves  the  ring,  passes  accurately  over  the  same 
number  of  inches  on  the  scale  in  each  successive  second. 

235.  Fourthly,  moving  bodies  have  a  constant  tendency  to  pro- 
ceed in  right  lines.     In  nature  there  occur  indeed  but  few  exam- 
ples of  rectilinear  motion,  but  almost  every  moving  body  describes 
a  curve.     Thus,  the  heavenly  bodies  move  in  ellipses  ;  projectiles 
describe  parabolas  ;  or  if  their  direction  is  so  altered  by  a  resist- 
ing medium,  as  the  atmosphere,  that  their  path  is  no  longer  a 
parabola,  it  is  still  changed  to  some  other  curve  ;  and  a  ship 
sailing  across  the  ocean,  describes  a  curvilinear  path  on  the  sur- 
face of  the  earth.     The  waving  of  trees  and  plants,  the  courses 
of  rivers,  the  spouting  of  fluids,  the  motions  of  winds  and  waves, 
are   likewise  more  or  less  curvilinear.     Bodies  falling  toward 
the  earth  by  gravity,  present  almost  the  only  examples  we  ob- 
serve in  nature  of  a  motion  purely  rectilinear.     But  notwith- 
standing the  deviation  from  a  right  line,  observable  in  actual 
motions,  yet  we  find  there  is  always  some  extraneous  cause  in 
operation,  which  accounts  for  such  deviation. 

236.  In  consequence  of  this  tendency  of  moving  bodies  to  pro- 
ceed in  right  lines,  when  a  body  revolves  in  a  curve,  around 
some  center  of  motion,  it  constantly  tends  to  fly  off  in  a  straight 
line,  which  is  a  tangent  to  its  orbit.     This  is  called  the  centrifu- 
gal force.     (Art.  185'.)     A  stone  from  a  sling,  water  escaping 
from  the  periphery  of  a  revolving  wheel,  and  water  receding 
from  the  center  of  a  tumbler  or  pail  when  the  vessel  is  whirled, 
are  familiar  instances  of  the  tendency  of  bodies  when  revolving 
in  circles  to  fly  off  in  straight  lines.     The  action  of  the  centrifu- 

*  There  are  frequently  connected  with  the  machine  a  pendulum,  and  such  parts  of 
a  clock  as  are  necessary  for  beating  seconds. 

23 


178  NATURAL   PHILOSOPHY. 

gal  force  may  be  studied  more  advantageously  by  means  of  an 
apparatus  called  the  Whirling  Tables.  These  consist  of  two 
small  circular  tables,  to  which  is  communicated  a  horizontal 
revolution  around  their  centers.  Bodies  laid  on  the  tables  in 
different  ways  are  made  to  participate  in  their  rotary  motions, 
and  thus  the  laws  of  the  centrifugal  force  may  be  observed. 
By  means  of  this  apparatus  the  following  propositions  are  illus- 
trated. 

237.  The  centrifugal  force  of  bodies  revolving  in  a  given  cir- 
cle, is  proportional  to  their  densities  or  specific  gravities.      If 
quicksilver,  water,  and  cork,  be  whirled  together  in  a  suspended 
pail  or  glass  vessel,  these  bodies  will  arrange  themselves  in  the 
order  of  their  specific  gravities,  so  that  the  cork  will  be  at  the 
least,  and  the  quicksilver  at  the  greatest  distance  from  the  center 
of  motion. 

238.  When  bodies  revolve  in  the  same  circle  with  different 
velocities,  the  centrifugal  forces  are  as  the  squares  of  the  veloci- 
ties.    By  doubling  the  velocity  of  a  revolving  body,  its  centrifu- 
gal  force  is  quadrupled.       Millstones,  revolving  horizontally, 
communicate  their  circular  motion  to  the  corn  that  is  introduced 
between  them  near  the  center.     The  corn,  by  the  centrifugal 
force  which  it  gradually  acquires,  recedes  from  the  center  and 
passes  out  at  the  circumference.     If  too  great  velocity  is  given 
to  millstones,  they  sometimes  burst  with  violence.     A  horse  in 
swift  motion,  on  suddenly  turning  a  corner,  throws  his  rider ; 
and  a  carriage  turning  swiftly  is  overset  on  the  same  principle. 
In  feats  of  horsemanship,  when  the  equestrian  rides  rapidly 
round  a  small  ring,  he  inclines  his  body  inward  in  different  de- 
grees, according  to  the  velocity  with  which  he  is  moving,  and 
thus  counteracts  his  tendency  to  fall  outward  by  the  centrifugal 
force.* 

239.  When  bodies  revolve  in  different  circles,  in  the  same  time, 
the  centrifugal  forces  are  as  the  radii  of  the  circles.     Hence, 
when  spherical  bodies  revolve  on  their  axes,  the  equatorial  parts, 
being  farthest  from  their  centers  of  motion,  and  consequently  mov- 
ing faster,  have  a  proportionally  greater  centrifugal  force.     If  the 
revolving  body  is  soft  so  as  to  yield,  it  is  elevated  in  the  equato- 
rial and  depressed  in  the  polar  parts.     Thus  a  mass  of  clay  re- 
volving on  a  potter's  wheel,  swells  out  in  the  central  parts  and 
becomes  flattened  at  the  two  ends.     The  earth  itself,  by  its  fig- 
ure, which  is  an  oblate  spheroid,  indicates  the  operation  of  this 
principle,  its  equatorial  exceeding  its  polar  diameter  by  26  miles ; 
and  the  planet  Saturn,  which  has  a  far  more  rapid  revolution  on 

*  Arnott's  El.  Phys.  p.  62. 


MECHANICS. 


179 


its  axis,  indicates  the  same  modification  of  its  figure  in  a  still 
higher  degree,  being  strikingly  elevated  at  the  equator  and  de- 
pressed at  the  poles. 

240.  The  centrifugal  force  Fie-  13L 

is  continually  opposed  to  the 
action  of  gravity,  so  that,  ex- 
cept at  the  pole,  where  this 
force  becomes  nothing,  the 
weight  of  a  body  is  diminished 
by  it  in  the  ratio  of  the  square 
of  the  cosine  of  the  latitude, 
For,  let  EPP  represent  the 
earth,  EQ  the  equator,  PP 
the  earth's  axis,  and  MN  the 
radius  of  a  parallel  of  latitude, 
which  equals  the  cosine  of  the 
latitude,  since  EM  is  the  lati- 
tude, MP  its  complement,  and  MN  the  sine  of  MP.  Now,  on 
account  of  the  different  velocities  of  two  bodies  at  E  and  M  re- 
spectively, their  centrifugal  forces  would  be  as  EO  to  MN,  or 
as  radius  to  the  cosine  of  the  latitude.  But,  at  the  equator,  the 
centrifugal  force  being  exerted  in  the  direction  of  OE  is  directly 
opposed  to  gravity,  while  at  M  the  force  is  exerted  in  the  direc- 
tion of  NM,  and  is  therefore  not  directly  opposed  to  gravity. 
Produce  MN  to  A,  and  let  MA  represent  the  centrifugal  force 
at  M,  and  upon  OM  produced  let  fall  the  perpendicular  AB ; 
then  MB  will  represent  the  part  of  the  force  which  acts  in  oppo- 
sition to  gravity.  Bui  since  the  angle  AMB  is  equal  to  the  lati- 
tude, (being  equal  to  MOE,)  therefore,  MA  is  to  MB  as  radius 
to  the  cosine  of  the  latitude.  Hence,  the  centrifugal  force  is  di- 
minished twice  in  the  ratio  of  the  cosine  of  the  latitude,  and 
consequently  is  as  the  square  of  the  cosine  of  the  latitude.  At 
the  equator,  the  centrifugal  force  is  ^  of  the  force  of  gravity, 
and  of  course  it  so  much  diminishes  the  weight  of  bodies  ;  and 
since  this  force  varies  as  the  square  of  the  velocity,  were  the 
earth  to  revolve  with  17  times  its  present  velocity  (the  square 
of  which  is  289)  the  centrifugal  force  would  be  equal  to  that  of 
gravity,  and  bodies  would  lose  all  their  weight ;  and  were  the 
velocity  greater  than  this,  they  would  fly  off  in  the  direction  of 
tangents  to  the  surface.* 

241.  We  have  hitherto  considered  the  doctrine  of  central  forces 
only  in  respect  to  bodies  moving  in  circular  orbits  ;  but  Newton 
extended  the  theory  to  all  possible  curves,  and  established  the 


*  At  the  equator,  the  loss  of  weight  by  the  centrifugal   force  is  -j^g- ;  and  by  in- 
creased distance  from  the  center  of  the  earth,  -g^o  ;  hence  the  entire  loss  is  xtr- 


180  NATURAL   PHILOSOPHY. 

general  laws  of  motion  in  every  sort  of  curve.  Observing  that 
we  can  always  make  a  circle  pass  through  three  points  taken  in 
any  curve,  and  the  nearer  the  points  are  to  each  other,  the  more 
nearly  the  circumference  of  the  circle  will  coincide  with  the  curve. 
If  the  three  points  be  taken  infinitely  near  each  other,  the  circle 
will  form  what  is  called  the  osculating  circle.  We  can  then 
suppose  that  at  each  point  of  the  curve  the  body  is  moving  in 
the  osculating  circle  ;  and,  consequently,  its  centrifugal  force  is 
measured  by  the  square  of  the  velocity  divided  by  the  radius  of  the 
osculating  circle,  (Art.  185'.)  As  the  radius  changes  at  each  point 
of  the  supposed  curve,  the  centrifugal  force  also  is  continually 
changing,  while  in  a  circular  orbit  it  remains  always  the  same.* 

242.  The  consideration  of  centrifugal  force  proves,  that  if  a 
body  be  observed  to  move  in  a  curvilinear  path,  some  efficient 
cause  must  exist  which  prevents  it  from  flying  off,  and  which 
compels  it  to  revolve  round  the  center.     Thus  the  bodies  of  the 
solar  system  are  constantly  impelled  or  drawn  toward  the  sun 
by  a  force  which  we  denominate  gravity.     If  this  force  did  not 
act  constantly,  they  would  resume  their  motion  in  the  right  line 
in  which  they  were  originally  projected,  when  they  were  first 
launched  into  space,  and  continue  moving  in  it  forever. 

243.  SECOND  LAW. — Motion,  or  change   of  motion,  is  propor- 
tional to  the  force  impressed,  and  is  produced  in  the  right  line  in 
which  that  force  acts. 

First,  motion  is  proportional  to  the  force  impressed.  This  is 
very  satisfactorily  shown  by  means  of  Atwood's  machine,  (Art. 
223,  Fig.  130.)  When  the  box  D  is  loaded  with  small  bars  of 
different  weights,  (the  bars  being  left  on  the  ring  H,  as  in  Art. 
223,)  the  box  descends  along  the  scale,  in  consequence  of  the 
motion  given  it  by  the  bars,  with  velocities  exactly  proportional 
to  the  weights  of  the  bars  respectively. 

Secondly,  motion  is  in  the  direction  of  the  force  impressed. 
Notwithstanding  the  diversity  of  motions  to  which  every  terres- 
trial body  is  constantly  subject,  the  effect  of  any  force  to  produce 
motion  is  the  same,  when  the  spectator  has  the  same  motion  as 
the  body,  as  though  the  body  were  absolutely  at  rest.  In  other 
words,  all  motions  are  compounded  so  as  not  to  disturb  each 
other  ;  each  remaining,  relatively,  the  same  as  if  there  were  no 
others.f  Since,  for  example,  by  the  diurnal  motion  of  the  earth, 
places  toward  the  equator  move  faster  than  those  toward  the 
poles,  if  the  foregoing  principle  were  not  true,  the  same  forces 
would  produce  different  quantities  of  motion  in  different  lati- 
tudes ;  and  a  body  struck  in  a  direction  north  or  south,  would 
not  move  in  that-  direction,  but  would  deviate  to  the  east  or  west. 

*  Pontecoulant,  El.  Astron.  p.  502.  t  Whewell's  Mechanics,  p.  231. 


MECHANICS.  181 

A  pendulum  also  would  vibrate  differently  according  as  it  moved 
in  a  north  and  south,  or  in  an  east  and  west  direction,  whereas 
not  the  slightest  difference  of  time  can  now  be  detected.  If  we 
are  in  a  ship,  moving  equally,  any  force  which  we  can  exert  will 
produce  the  same  motion  relatively  to  the  vessel,  whether  it  be  or 
be  not  in  the  direction  of  the  vessel's  motion.  If  we  stand  on 
the  deck,  supposed  to  be  level,  and  roll  a  body  along  it,  the  same 
effort  will  produce  the  same  velocity  along  the  deck  whether  the 
motion  be  from  head  to  stern,  or  from  stem  to  head,  or  across  the 
vessel.  Also,  a  body  dropped  from  the  top  of  the  mast  will  not 
be  left  behind  by  the  motion  of  the  ship,  but  will  fall  along  the 
mast  as  it  would  if  the  mast  were  at  rest,  and  will  reach  the  foot 
of  it  in  the  same  time.  If  a  body  be  thrown  perpendicularly 
upward,  it  will  rise  directly  over  the  hand  and  fall  perpendicu- 
larly upon  it  again ;  and  if  it  be  thrown  in  any  other  direction, 
the  path  and  motion  relatively  to  the  person  who  throws  it  will  be 
the  same  as  if  he  were  at  rest.* 

244.  It  may  seem,  at  first  view,  more  questionable  whether,  as 
is  asserted  in  Art.  20,  the  smallest  force  is  capable  of  moving  the 
largest  body.     Agreeably  to  this  doctrine,  a  blow  with  a  hammer 
upon  the  earth  ought  to  move  it,  and  that  it  would  do  so  may  be 
inferred  from  the  following  reasons. 

(1.)  We  can  conceive  the  earth  to  be  divided  into  parts  so 
small,  that  the  blow  would  produce  upon  one  of  them  even  a 
sensible  motion.  Then  it  would  produce  on  two  of  the  parts 
half  as  much  velocity  ;  and  upon  all  the  parts  together  a  velocity 
as  much  less  than  upon  one,  as  their  number  was  greater  than 
unity.  This  velocity  might  be  appreciable  in  numbers,  although 
too  small  to  be  observed  by  the  senses. 

(2.)  Very  heavy  weights  may  be  actually  put  in  motion  by 
small  forces.  Leslie  asserts  that  a  ship  of  any  burden  in  calm 
weather  and  smooth  water,  may  be  gradually  pulled  along  even 
by  the  exertions  of  a  boy.f 

(3.)  The  repetition  of  very  small  blows  finally  produces  sen- 
sible effects  upon  large  bodies.  The  wearing  away  of  stone  by 
the  dropping  of  water,  the  abrasion  of  marble  images  by  the  kisses 
of  pilgrims,  and,  especially,  the  demolition  of  the  strongest  for- 
tresses by  repeated  blows  of  the  battering  ram,  are  examples  of 
powerful  effects  produced  by  small  impulses,  each  of  which  must 
have  contributed  its  share,  since  the  addition  of  any  number  of 
nothings  is  nothing  still. 

245.  THIRD  LAW. — When  bodies  act  upon  each  other,  action  and 
reaction  are  equal,  and  in  opposite  directions. 

*  Robison's  Mech.  Phil,  by  Brewster,  I,  42. 
t  Leslie,  El.  Nat.  Phil.  I,  30. 


182  NATURAL    PHILOSOPHY. 

The  doctrine  of  action  and  reac^on  has  been  fully  investigated 
and  explained  in  the  former  part  of  this  work.  All  we  propose 
to  do  at  present  is  to  add  a  few  familiar  and  practical  illus- 
trations. 

246.  If  I  strike  one  hand  upon  the  other  at  rest,  I  perceive  no 
difference  in  the  sensations  experienced  by  each.    The  resistance 
to  the  hand  which  gives  the  blow  is  equal  to  the  impulse  given. 
A  boatman  presses  against  the  bank  with  his  oar,  and  receives 
motion  in  the  opposite  direction,  which  being  communicated  to 
the  boat,  makes  it  recede  from  the  shore.     He  strikes  the  water, 
the  reaction  of  which  at  every  impulse,  carries  the  boat  forward 
in  the  opposite  direction.     An  infirm  old  man  presses  the  ground 
with  his  staff,  and  thus,  by  lightening  the  pressure  on  his  lower 
limbs,  makes  his  arms  perform  a  part  of  the  labor  of  walking. 
A  bird  beats  the  air  with  his  wings,  and  by  giving  a    blow 
whose  reaction  is  more  than  sufficient  to  balance  the  weight  of 
his  body,  rises  with  the  difference.     When  the  wings  are  small 
and  slender,  as  those  of  the  humming-bird,  and  disproportioned 
to  the  weight  of  the  body,  the  defect  is  compensated  by  more 
frequent  blows,  giving  nimble  motions,  suited  to  their  short  but 
swift  excursions,  while  the  long  wings  of  the  eagle  are  equally 
fitted,  by  their  less  rapid  but  more  effectual  blows,  for  his  distant 
journeys  through  the  skies.     Hence,  propelling  and  rowing  a 
boat,  flying,  and  swimming,  are  processes  analogous  to  each 
other,  depending  on  the  principle  of  reaction. 

247.  If  a  man  stands  in  a  boat  and  pulls  upon  a  rope  which  is 
fastened  to  a  post  on  the  shore,  the  force  of  the  man  is  expended 
on  the  post  in  one  direction,  and  the  post,  by  its  reaction,  draws 
the  man  in  the  opposite  direction,  namely,  toward  the  shore. 
(See  p.  32,  Ex.  12.)     Call  the  man  A,  and  let  another  man  B 
take  the  place  of  the  post.     If  B  pulls  with  a  force  just  equal  to 
that  of  A,  he  will  do  nothing  more  than  what  the  post  did  before, 
and  therefore  the  two  men  together  will  bring  the  boat  ashore 
no  sooner  than  A  would  have  done  alone  in  the  former  case.    If 
A  pulls  with  more  force  than  B,  he  pulls  B  toward  him,  and  the 
reaction,  or  the  force  which  carries  the  boat  ashore,  is  the  same 
as  before,  namely,  the  force  of  B.     If  B  were  to  pull  with  more 
force  than  A,  he  would  pull  A  out  of  the  boat,  were  not  A  at- 
tached firmly  to  the  boat,  in  which  case  the  velocity  of  the  boat 
would  be  augmented.     By  attentively  considering  this  and  all 
analogous  cases,  we  shall  perceive,  that  whenever  two  bodies 
act  against  each  other,  they  give  and  receive  equal  momenta, 
and  the  momenta  being  in  opposite  directions,  it  follows,  that 
bodies  do  not  alter  the  quantity  of  motion  they  have,  estimated 
in  a  given  direction,  by  their  mutual  action  on  each  other.    This 
principle  is  well  explained  in  Emerson's  Mechanics,  as  follows : 


MECHANICS.  183 

The  sum  of  the  motions  of  any  two  bodies  in  any  one  line  of  di- 
rection, toward  the  same  part,  cannot  be  changed,  by  any  action  of 


in  one  and  the  same  line  of  direction,  and  always  the  same  way, 
is  eternally  and  invariably  the  same.  Whatever  motion,  there- 
fore, one  body  receives  toward  another,  whether  it  is  drawn  to- 
ward it  by  attf  action,  or  by  a  rope,  or  by  any  other  method,  pre- 
cisely the  same  quantity  of  motion  it  imparts  to  the  other  body 
in  the  opposite  direction-!  If  a  man  in  a  boat  pulls  at  a  rope 
attached  .to  another  boat  of  equal  size,  the  boats  will  move  to- 
ward each  other  with  equal  velocities ;  but  a  man  in  a  boat 
pulling  a  rope  attached  to  a  large  ship  seems  only  to  move  the 
boat,  but  he  really  moves  the  ship  a  little,  although  its  velocity 
is  as  much  less  than  that  of  the  boat,  as  its  weight  is  greater. 
A  pound  of  lead  and  the  earth  attract  each  other  with  equal 
force,  and  the  two  bodies  approach  each  other  with  equal  mo- 
ment a.  J 

248.  Since  momentum  is  proportioned  to  the  joint  product;  of 
the  velocity  and  quantity  of  matter,  a  great  momentum  may  be 
obtained  either  by  giving  a  slow  motion  to  a  great  mass,  or  a 
swift  motion  to  a  small  body.     A  striking  illustration  of  this  is 
afforded  by  problem  9th,  page  32,  where  on  the  supposition 
that  a  grain  of  light  moving  with  its  usual  velocity,  were  to  im- 
pinge directly  against  a  mass  of  ice  floating  at  its  ordinary  slow 
rate,  the  grain  of  light  would  be  competent  to  stop  about  44^ 
tons  of  ice.     Islands  of  ice  move  with  such  vast  momentum,  that 
they  instantly  demolish  the  largest  ship  of  war  if  it  comes  in 
their  vVay. 

249.  If  a  body  in  motion  strikes  a  body  at  rest,  the  striking 
body  must  sustain  as  great  a  shock  from  the  collision  as  if  it  had 
been  at  rest,  and  struck  by  the  other  body  with  the  same  force. 
For  the  loss  of  force  which  it  sustains  in  one  direction  is  an 
effect  of  the  same  kind  as  if,  being  at  rest,  it  had  received  as 
much  force  in  the  opposite  direction.     If  a  man  walking  rapidly, 
or  running,  encounters  another  standing  still,  he  suffers  as  much 
from  the  collision  as  the  man  against  whom  he  strikes.     When 
two  bodies  moving  in  opposite  directions  meet,  each  body  sus- 
tains as  great  a  shock  as  if,  being  at  rest,  it  had  been  struck  by 

*  Emerson's  Mechanics,  4to,  p.  17. 

t  Quantitas  motus  quae  colligitur  capiendo  suramam  motuum  factorum  ad  ean- 
dem  partem,  et  difFerentiam  factorum  ad  contrarias,  non  mutatur  ab  actioiie  corpo- 
rum  inter  se.  (Principia,  Lex  III,  cor.  3.) 

t  The  pound  of  lead  does  indeed  attract  the  earth  only  half  as  much  as  two 
pounds  would  do  ;  nor  does  it  receive  from  the  earth  but  half  as  much  ;  the  power  of 
attracting  and  of  being  attracted  is  the  same. 


184  NATURAL   PHILOSOPHY. 

the  other  body  with  the  united  forces  of  the  two.  For  this  rea- 
son, two  persons  walking  in  opposite  directions,  receive  from 
their  encounter  a  more  violent  shock  than  might  be  expected. 
If  they  be  of  nearly  equal  weight,  and  one  be  walking  at  the 
rate  of  three  and  the  other  four  miles  an  hour,  each  sustains  the 
same  shock  as  if  he  had  been  at  rest,  and  struck  by  the  other 
running  at  the  rate  of  seven  miles  an  hour.* 

This  principle  accounts  for  the  destructive  effects  arising  from 
ships  running  foul  of  each  other  at  sea.  If  t\tfo  ships  of  500 
tons  burden  encounter  each  other,  sailing  ten  knots  an  hour, 
each  sustains  the  shock  which  being  at  rest  it  would  receive 
from  a  vessel  of  1000  tons  burden  sailing  at  the  same  rate. 
It  is  a  mistake  to  suppose  that  when  a  large  and  a  small  body 
meet,  the  small  body  suffers  a  greater  shock  than  the  large  one. 
The  shock  which  they  sustain  must  be  the  same  ;  but  the  large 
body  may  be  better  able  to  bear  it.  When  the  fist  of  a  pugilist 
strikes  the  body  of  his  antagonist,  it  sustains  as  great  a  shock  as 
it  gives  ;  but  the  part  being  more  fitted  to  endure  the  blow,  the 
injury  and  pain  are  inflicted  on  his  opponent.  This  is  not  the 
case  however  when  fist  meets  fist.  Then  the  parts  in  collision 
are  equally  sensitive  and  vulnerable,  and  the  effect  is  aggravated 
by  both  having  approached  each  other  with  great  force.  The 
effect  of  the  blow  is  the  same  as  if  one  fist,  being  held  at  rest 
were  struck  by  the  other  with  the  combined  force  of  both.f 

250.  The  question  maybe  asked,  Why  are  the  effects  so  much 
more  severe  when  we  fall  from  an  eminence  upon  a  naked  rock 
than  upon  a  bed  of  down  ?    In  both  instances  our  fall  is  arrested, 
and  we  sustain  a  contrary  and  equal  reaction  ;  yet  in  the  one 
case  we  might  suffer  hardly  any  injury,  while  in  the  other  we 
should  be  bruised  to  death.     The  reason  of  the  difference  is  this : 
when  we  fall  on  a  bed  of  down,  the  resistance  is  applied  gradu- 
ally ;  when  we  fall  on  a  rock,  it  is  applied  instantaneously.    We 
do  not  strike  the  bed  with  the  same  force  that  we  do  the  rock  ; 
we  move  along  with  the  bed,  and  of  course  do  not  lose  our  mo- 
tion at  once,  and  we  receive  in  the  opposite  direction  merely 
what  we  lose.     A  violent  blow,  if  equally  diffused  over  the  hu- 
man body,  may  be  sustained  without  injury.     Thus,  if  an  anvil 
be  laid  on  the  breast,  a  man  may  receive  on  it  a  heavy  blow  from 
a  great  hammer  with  impunity.  J 

251.  There  are  many  instances  where  action  and  reaction 
mutually  destroy  each  other  and  no  motion  results.     Thus,  when 
a  child  stands  in  a  boat  and  pulls  by  a  rope  attached  to  the  stern, 
he  labors  in  vain  to  make  the  boat  advance.     Dr.  Arnott  tells  us 
of  a  man  who  attached  a  large  bellows  to  the  hinder  part  of  his 

*  Lardner's  Mech.,  p.  47  t  Ib.  p.  48.  t  Araott's  El.  Phys.  p.  104. 


MECHANICS.  185 

boat,  with  the  view  of  manufacturing  a  breeze  for  himself,  being 
ignorant  that  the  reaction  would  carry  the  boat  backward  as 
much  as  the  impulse  of  the  artificial  wind  carried  it  forward.* 
A  force  which  begins  and  ends  within  a  machine  has  no  power 
to  move  it.f 

252.  VARIABLE  MOTION. — When  a  moving  body  is  subjected  to 
the  energy  of  a  force  which  acts  on  it  without  interruption,  but 
in  a  different  manner  at  each  instant,  the  motion  is  called  in 
general,  variable  motion.^     We  have  instances  of  variable  mo- 
tions in  the  unbending  of  springs,  in  the  action  of  the  wind  on 
the  sails  of  a  ship,  and  in  the  action  of  gunpowder  on  a  ball 
while  it  is  passing  through  the  barrel  of  the  gun.     In  each  of 
these  cases,  the  velocity  of  the  moving  body  is  constantly  aug- 
mented, yet  the  degree  of  augmentation  is  diminishing  until  it 
finally  ceases. 

253.  When  a  moving  body  receives,  each  successive  instant, 
the  same  increase  of  velocity,  it  is  said  to  be  uniformly  accelerated. 
If  a  small  wheel  were  revolving  without  resistance,  and  at  the 
end  of  every  second  I  should  apply  a  given  impulse,  the  wheel 
would  be  uniformly  accelerated  ;  for,  by  its  own  inertia,  it  would 
retain  all  its  previous  motion,  and,  by  the  second  law  of  motion, 
the  repetition  of  the  same  force,  at  equal  intervals,  would  increase 
its  velocity  at  a  uniform  rate.     If  the  intervals  at  which  this 
force  was  repeated,  were  indefinitely  diminished,  the  same  kind 
of  effect  would  take  place ;  and  the  same  would  evidently  be 
the  case,  were  the  force  to  operate  without  cessation.     Such  a 
force  is  that  of  GRAVITY. 

254.  It  has  already  been  shown,  in   articles  4  and  7,  that 
Gravity  is  a  quality  which  belongs  alike  to  all  matter  in  propor- 
tion to  its  quantity ;  and  that  at  different  distances  from  the 
center  of  the  earth,  it  varies  inversely  as  the  square  of  the  dis- 
tance.    The  manner  in  which  this  force  decreases  as  the  distance 
increases  will  be  seen  at  one  view  by  the  following  table,  begin- 
ning with  the  distance  of  the  surface  from  the  center. 

*  Arnott's  El.  Phys.  p.  107. 

t  It  is  common  in  elementary  works  on  Mechanics,  to  find  under  the  head  of  "  re- 
action,"  a  class  of  phenomena  which  evidently  belong  to  a  cause  distinct  from  that 
of  the  mutual  action  of  bodies.  For  example,  a  little  steam  carriage  is  sometimes 
exhibited,  from  which  a  jet  of  steam  issues,  and  the  carriage  moves  in  the  opposite 
direction.  This,  it  is  said,  is  owing  to  the  reaction  of  the  air  upon  the  steam,  being 
supposed  analogous  to  the  flying  of  a  bird,  which  beats  the  air  with  its  wings,  and  is 
borne  along  by  its  reaction ;  but  the  motion  of  the  carriage  in  the  foregoing  experi- 
ment, is  owing  to  a  very  different  cause.  Before  the  jet  was  opened,  the  steam 
pressed  equally  on  all  sides  of  the  vessel ;  as  soon  as  the  opening  is  made  on  one  side, 
the  pressure  is  removed  from  that  side,  but  remains  on  the  opposite  side,  and  therefor* 
gives  motion  to  the  vessel  in  that  direction. 

t  Gregory's  Mech.,  I,  81. 

24 


186 


NATURAL   PHILOSOPHY 


Distance, 

1 

2 

3 

4 

5 

10 

Tloo" 

20 

To'o" 

60 

Attraction, 

1 

i 

7 

* 

A 

Tl 

?eV^ 

Hence  it  appears  that  a  body  placed  20  times  as  far  from  the 
center  of  the  earth  as  the  surface  is  from  the  center,  is  attracted 
only  ?^th  part  as  much;  and  at  the  distance  of  60  times  the 
radius  of  the  earth  the  same  force  is  diminished  3600  times.* 
At  this  distance  therefore  it  would  take  60  seconds,  or  one  minute, 
for  a  body  to  fall  through  the  space  it  falls  at  the  surface  of  the 
earth  in  one  second ;  that  is,  through  16^  feet.  But  all  dis- 
tances within  a  few  hundred  feet  of  the  earth  bear  so  small  a 
ratio  to  the  earth's  radius,  that  the  force  of  gravity  may  be  con- 
sidered as  the  same  unvarying  force,  in  relation  both  to  the 
weights  of  bodies  and  to  the  velocities  with  which  they  fall. 
(See  Art.  8.) 

255.  It  is  not  alone  by  the  direct  fall  of  bodies  that  the,  gravi- 
tation of  the  earth  is  manifested.  The  curvilinear  motion  of 
bodies  projected  in  directions  different  from  the  perpendicular,  is 
a  combination  of  the  effects  of  the  uniform  velocity  that  has 
been  given  to  the  projectile  by  the  impulse  which  it  has  received, 
and  the  accelerated  or  retarded  velocity  which  it  receives  from 
the  earth's  attraction.  Suppose  a  Fig. 

body  placed  at  any  point  P,  (Fig. 
132,)  above  the  surface  of  the  earth, 
and  let  PA  be  the  direction  of  the 
earth's  center.  If  the  body  were 
allowed  to  move  without  receiving 
any  impulse,  it  would  descend  to 
the  earth  in  the  direction  PA  with 
an  accelerated  motion.  But  sup- 
pose that  at  the  moment  of  its  de- 
parture from  P,  it  receives  an  im- 
pulse in  the  direction  PB,  which 
would  carry  it  to  B  in  the  time  the 
body  would  fall  from  P  to  A ;  then, 
by  the  composition  of  motion,  (Art.  41,)  the  body  must  at  the  end 
of  that  time  be  found  in  the  line  BD,  parallel  to  PA.  If  the  mo- 
tion in  the  direction  of  PA  were  uniform,  the  body  P  would  in 
this  case  move  in  the  straight  line  PD.  But  this  is  not  the  case. 
The  velocity  of  the  body  in  the  direction  PA  is  at  first  so  small 
as  to  produce  very  little  deflection  of  its  motion  from  the  line  PB. 
As  the  velocity,  however,  increases,  this  deflection  increases,  so 
that  it  moves  Jrom  P  to  D  in  a  curve,  which  is  convex  toward 
PB.  The  greater  the  velocity  of  the  projectile  in  the  direction 

*  This  last  is  nearly  the  distance  of  the  moon  from  the  earth  ;  and  it  is  found  by 
calculation  that  the  moon  is  actually  drawn  toward  the  earth,  away  from  the  straight 
line  in  which  she  tends  to  move,  by  exactly  this  force. 


MECHANICS.  187 

PB,  the  greater  sweep  the  curve  will  take.  Thus  it  will  succes- 
sively take  the  forms  PD,  PE,  PF,  &c.,  and  that  velocity  can  be 
computed,*  which  (setting  aside  the  resistance  of  the  air)  would 
cause  the  projectile  to  go  completely  round  the  earth,  and  return 
to  the  point  P  from  which  it  departed.  In  this  case  the  body  P 
would  continue  to  revolve  round  the  earth  like  the  moon. 

256.  Hence  it  is  obvious,  that  the  phenomenon  of  the  revolu- 
tion of  the  moon  round  the  earth,  is  nothing  more  than  the  com- 
bined effects  of  the  earth's  attraction,  and  the  tendency  it  has  to 
move  forward  in  a  straight  line  which  is  a  tangent  to  its  orbit. 
And  were  any  of  the  heavenly  bodies  to  explode,  we  may  con- 
ceive that  the  fragments  would  proceed  in  a  rectilineal  direction 
until,  approaching,  severally,  within  the  sphere  of  influence  of  some 
large  body,  whose  attraction  would  combine  with  their  projectile 
force,  they  would  forever  afterward  continue  to  revolve  around 
that  body,  as  the  satellites  revolve  around  their  primaries.f 

257.  But  the  attraction  of  gravitation  is  manifested  by  com- 
paratively small  masses  of  matter.     The  effect  of  a  high  moun- 
tain is  perceptible  upon  a  plumb  line,  causing  it  to  deviate 
sensibly  from  a  perpendicular,  so  that  the  same  star  in  the  zenith 
would  change  its  apparent  place  when  viewed  on  opposite  sides 
of  the  mountain.     This  was  observed  by  two  French  astrono- 
mers, near  Mount  Chimborazo  in  South  America,  as  early  as  the 
year  1738 ;  and  the  experiment  was  repeated  in  1772  with  all 
possible  accuracy,  by  Dr.  Maskelyne,  astronomer  royal  of  Great 
Britain,  at  the  base  of  the  mountain  Schehallien,  in  the  eastern 
part  of  Scotland.;};     Mr.  Cavendish,   a  distinguished   English 
philosopher  of  the  last  century,  rendered  sensible  even  the  attrac- 
tion of  a  sphere  of  lead,  by  bringing  it  near  a  small  bullet,  sus- 
pended from  one  arm  of  an  exceedingly  delicate  balance.     The 
sphere  when  brought  near  the  bullet  disturbed  its  equilibrium. 

258.  By  gravity,  bodies  are  directed  toward  the  center  of  the 
earth.     We  are  not  to  infer  from  this  fact  that  there  is  any  pecu- 
liar force,  (like  that  of  a  large  magnet,  for  example,)  residing  at 
the  center,  but  merely  that  the  effect  of  the  earth,  taken  as  a 
whole,  is  the  same  as  though  its  matter  were  condensed  into  the 
center.    The  line  of  attraction  passes  through  the  center  because 

*  A  cannon  ball  shot  horizontally  from  the  top  of  a  lofty  mountain,  would  go  three 
or  four  miles.  If  there  were  no  atmosphere  to  resist  its  motion,  the  same  original 
velocity  would  carry  it  thirty  or  forty  miles  before  it  fell ;  and  if  it  could  be  dispatched 
with  about  ten  times  the  velocity  of  a  cannon  shot,  the  centrifugal  force  would  ex- 
actly balance  the  force  of  gravity,  and  the  ball  would  go  quite  round  the  earth.  (Ar- 
uott,  El.  Phys.  p.  91.) 

t  This  has  actually  been  supposed  of  the  four  new  planets,  Ceres,  Pallas,  Juno,  and 
Vesta. 

t  Ed.  Encyclopedia,  III,  76. 


188 


NATURAL   PHILOSOPHY. 


such  a  line  is  the  resultant  of  the  separate  attractions  of  all  the 
particles  composing  a  sphere.  Thus  if  AB  be  a  line  passing 
through  the  center  of  the  sphere,  and  A  be  FlS- 133- 

any  body,  and  a  and  b  two  particles  of  mat- 
ter  in  the  sphere  equally  distant  from  A,  it  is 
evident  that  their  combined  actions  would 
be  expressed  by  the  diagonal  which  would 
coincide  in  direction  with  AB.  The  same 
is  true  of  any  other  particles  taken  equally 
distant  from  A  in  opposite  hemispheres.  At 
different  parts  of  the  earth,  therefore,  the  di- 
rections of  falling  bodies  are  not  parallel, 
but  form  converging  lines.  But  on  account 
of  the  great  magnitude  of  the  earth,  two 
places  100  feet  distant  will  not  vary  one 
second,  and  when  a  mile  asunder,  they  will  not  differ  one  minute 
from  perfect  parallelism.*  For  all  the  purposes  of  machinery, 
therefore,  as  well  as  for  experiment,  the  direction  of  the  lines  of 
gravity  may  be  considered  as  parallel. 

259.  Since  bodies  in  falling  toward  the  earth  are  uniformly 
accelerated,  the  velocity  acquired  must  be  proportioned  to  the  time 
the  body  has  been  falling  :  at  the  end  of  ten  seconds  it  has  ac- 
quired ten  times  the  velocity  which  it  had  at  the  end  of  one 
second.     And  in  Art.  29,  it  has  been  shown  that  the  spaces  de- 
scribed are  proportioned  to  the  squares  of  the  times  ;  so  that  the 
space  described  during  100  seconds,  is  not  merely  100  times  as 
great  as  that  described  in  one  second,  but  it  is  the  square  of  100, 
or  10,000  times  as  great.     This  conclusion  was  arrived  at  mathe- 
matically, long  before  it  was  established  by  actual  experiment. 
There  were  two  difficulties  which  stood  in  the  way  of  such  a 
verification,  viz.,  the  little  time  occupied  in  descending  through 
such  perpendicular  heights  as  the  experimenter  can  command, 
and  the  resistance  of  the  air,  which,  when  the  velocity  becomes 
great,  acts  as  a  powerfully  retarding  force.     We  can  rarely  com- 
mand a  perpendicular  eminence  of  more  than  four  hundred  feet, 
and  yet  the  time  occupied  in  the  whole  descent  is  only  about  five 
seconds,  a  period  too  short  to  enable  us  to  mark  distinctly  the 
respective  rates  at  which  the  successive  intervals  are  described. 
Atwood's  machine  (Fig.  130,  Art.  231)  affords  the  means  of  obvia- 
ting both  these  difficulties,  and  verifying  the  laws  of  falling  bodies 
with  great  accuracy. 

260.  The  object  of  the  machine,  so  far  as  respects  experiments 
on  falling  bodies,  is  to  render  the  descent  of  bodies  so  gradual 
that  the  relation  between  the  times  and  spaces  can  be  accurately 


»  Gregory's  Mech.  I,  193. 


MECHANICS.  189 

observed.  By  rejurrence  to  the  figure,  and  to  the  descriptions 
given  in  Art.  231,  we  shall  readily  see  how  this  object  may  be 
accomplished.  The  weights  D  and  E  each  equal  31?  ounces, 
and  of  course  the  quantity  of  matter  in  both  is  63  ounces.  Now, 
since  one  of  these  rises  as  the  other  descends,  the  force  of  gravity 
retards  the  one  as  much  as  it  accelerates  the  other,  and  they  are 
in  effect  the  same  as  though  they  were  entirely  destitute  of  gravity. 
If  a  small  weight,  as  one  ounce,  were  let  fall  freely  from  the  top 
of  the  machine,  it  would  fall  through  so  small  a  space  almost  in 
an  instant,  and  we  should  be  unable  to  mark  the  rate  at  which  it 
would  pass  over  the  successive  portions  of  the  graduated  scale 
FG  ;  but  if  it  be  laid  on  the  weight  D,  it  must  carry  D  along 
with  it ;  that  is,  it  must  make  D  descend  and  E  ascend,  and 
therefore  the  motion  belonging  to  one  ounce,  will  be  distributed 
throughout  64  ounces,  and  its  velocity  will  be  retarded  in  the 
same  ratio.  Consequently  the  weight  D  will  descend  only  ^th 

Eart  as  fast  as  a  body  falling  freely ;  and  as  a  body  falling 
^eely  descends  in  one  second  about  16  feet  or   192  inches,  the 
weight  D  will  descend  l-/f  —  3  inches  in  one  second.     The  com- 
parative progress  of  this  weight,  and  of  a  body  falling  freely  for 
several  successive  seconds,  will  be  seen  in  the  following  table. 


Time, 

1 

2 

3 

4 

5 

6  ' 

Body  falling  freely,  in  feet, 

16r'o 
3 

641 

12 

144£ 

2571 

48 

402  T'5 

579 

Tos 

Do.  in  At  wood's  Machine,  in  inches, 

27 

75 

Hence  it  appears  that  in  six  seconds,  while  a  body  would  fall 
freely  through  579  feet,  it  would  in  the  same  time  descend  only 
nine  feet  in  Atwood's  Machine.  But  the  latter  is  a  uniformly 
accelerated  velocity,  and  subject  to  the  same  laws  as  the  former, 
and  it  may  therefore  be  employed  to  investigate  the  laws  of  fall- 
ing bodies.  The  results  correspond  remarkably  with  theory,  so 
that  when  the  instrument  is  well  constructed,  and  managed  skil- 
fully, the  descending  weight  clicks  upon  the  stage  or  brass  plate 
K,  at  the  very  instant  required. 

261.  We  see  in  nature  the  law  of  acceleration  of  falling  bod- 
ies indicated,  by  the  impetuosity  with  which  bodies  fall  from  any 
considerable  height  upon  the  earth.  Meteoric  stones,  falling  from 
the  sky,  sometimes  bury  themselves  deep  in  the  ground.  Aero- 
nauts that  have  fallen  from  balloons  have  been  dashed  in  pieces.* 

*  Any  liquid  falling  from  a  reservoir,  forms  a  descending  mass  or  stream,  of 
which  the  bulk  diminishes  from  above  downward,  in  the  same  proportion  in  which 
the  velocity  increases.  This  truth  is  well  exemplified  by  the  pouring  out  of  mo- 
lasses or  thick  sirup :  if  the  height  of  the  fall  be  considerable,  the  bulky  mass  which 
first  escapes,  is  reduced,  before  it  reaches  the  bottom,  to  a  small  thread ;  but  the 
thread  is  moving  with  proportionally  greater  speed,  for  it  fills  the  receiving  vessel 
with  great  rapidity.  The  same  truth  is  exhibited  on  a  grand  scale  in  the  Falls  of 
Niagara,  where  the  broad  river  is  seen  first  bending  over  the  precipice,  a  vast  slow 


190  NATURAL   PHILOSOPHY. 

It  is,  however,  a  rare  occurrence  to  see  a  body  falling  from  afry 
great  height  perpendicularly :  most  instances  of  accelerated  mo- 
tion which  come  under  our  observation,  are  in  bodies  falling 
down  inclined  planes,  where  the  same  law  of  acceleration  prevails. 
(Art.  158.)  A  fragment  of  rock  descending  from  the  side  of  a 
mountain,  has  its  speed  augmented  as  it  goes,  until  its  momen- 
tum becomes  irresistible,  and  large  trees  are  prostrated  before  it. 

262.  A  very  remarkable  example  of  the  acceleration  of  bodies 
descending  down  inclined  planes,  occurs  at  the  Slide  of  Alpnach 
in  Switzerland.    On  Mount  Pilatus.  near  Lake  Luzerne,  is  a  val- 
uable growth  of  fir-trees,  which,  on  account  of  the  inaccessible 
nature  of  the  mountain,  had  remained  for  ages  undisturbed,  until 
within  a  few  years,  a  German  engineer  contrived  to  construct  a 
trough  in  the  form  of  an  inclined  plane,  by  which  these  trees 
are  made  to  descend  by  their  own  weight,  through  a  space  of 
eight  or  nine  miles,  from  the  side  of  the  mountain  to  the  margin 
of  the  lake.     Although  the  average  declivity  is  no  more  than 
about  one  foot  in  seventeen,  and  the  route  often  circuitous  and 
sometimes  horizontal,  yet  so  great  is  the  acceleration,  that  a  tree 
descends  the  whole  distance  in  the  short  space  of  six  minutes. 
To  a  spectator  standing  by  the  side  of  the  trough,  at  first  is 
heard,  on  the  approach  of  the  tree,  a  roaring  noise,  becoming 
louder  and  louder ;  the  tree  comes  in  sight  at  the  distance  of  half 
a  mile,  and  in  an  instant  afterward  shoots  past  with  the  noise  of 
thunder  and  the  rapidity  of  lightning.     When  a  tree  happens  to 
"bolt"  from  the  trough,  it  cuts  the  standing  trees  quite  off.* 
(See  p.  131,  Prob.  5  and  6.) 

263.  COMPOSITION  AND  RESOLUTION  OF  MOTION. — Simple  motion 
is  that  which  arises  from  the  action  of  a  single  force  ;  Compound 
motion  is  that  which  is  produced  by  several  forces  acting  in  dif- 
ferent directions.     Strictly  speaking,  we  have  no  example  of  a 
simple  motion,  since  in  the  absolute  motion  of  all  bodies,  their 
own  proper  motion  is  combined  with  that  of  the  earth  in  its  di- 
urnal and  annual  revolutions,  and  we  know  not  with  how  many 
others.     (Art.  226.)     In  an  enlarged  sense,  therefore,  all  motions 
are  compound.     But  in  the  foregoing  distinctions  we  have  refe- 
rence only  to  relative  motions,  as  those  which  take  place  among 
bodies  on  the  earth.     In  accordance  with  the  second  law  of  mo- 
tion, (Art.  243,)  a  force  striking  upon  a  body  in  motion,  will  pro- 
duce the  same  change  of  motion  as  though  the  body  had  been  at 
rest  when  the  force  struck  it.    This  may  at  first  view  appear  in- 
consistent with  experience,  especially  in  regard  to  opposite  mo- 
moving  mass,  then  becoming  a  thinner  and  thinner  sheet ;  until  it  flashes  into  the 
deep  below,  almost  with  the  vdjcity  of  lightning.     (Arnott's  El.  Phys.  79.) 

*  Plavfair's  Works,  I,  96. 


MECHANICS.  191 

tions.  Let  us,  therefore,  consider  the  principle  in  its  application 
to  several  different  cases.  Conceive  the  ice  of  a  frozen  river  to 
be  first  stationary,  and  afterwards  to  float  down  with  the  current. 
Standing  on  the  bank  I  roll  a  ball  directly  across  the  river.  Will 
it  pass  in  the  same  direction  in  both  cases  ?  It  will  not ;  for  in 
the  first  case  it  will  pass  across  perpendicularly  to  the  banks,  and 
in  the  second  case  it  will  go  across  diagonally.  But  now  let  me 
stand  upon  the  ice  and  roll  the  ball.  Since  I  float  along  with 
the  ice,  I  am  at  rest  with  respect  to  that  motion,  and  the  ball, 
though  moving  diagonally  as  before,  appears  to  me  to  go  directly 
across  the  stream.*  If  the  ball  was  rolled  not  directly  across  but 
obliquely,  making  any  angle  with  the  bank,  if  I  stood  upon  the 
floating  ice,  and  was  therefore  at  rest  with  respect  to  one  of  the 
motions  of  the  ball,  I  should  see  the  other  motion  in  the  same 
manner  as  though  I  had  stood  on  the  shore  and  the  ice  had  been 
at  rest.  But  if  when  a  ball  is  rolling  towards  me,  and  I  strike  it 
in  a  direction  exactly  opposite  to  its  course,  but  do  not  stop  it, 
can  the  blow  be  said  to  produce  the  same  change  of  motion  as 
though  the  body  had  been  at  rest  when  the  blow  was  applied  ? 
Ans.  If  I  had  been  moving  in  the  same  manner  as  the  ball  before 
the  blow,  then  stopping  a  part  of  the  motion  of  the  ball,  would 
have  given  it  a  relative  motion  in  the  opposite  direction  ;  since 
having  none  of  my  own  motion  stopped,  I  should  leave  it  be- 
hind. This  is  what  takes  place  when  a  cannon  ball  is  fired  in  a 
direction  contrary  to  that  in  which  the  earth  is  revolving  about 
the  sun.  The  cannon  moves  onward  and  leaves  the  ball  behind. 
(Art.  226.) 

264.  The  laws  respecting  the  composition  and  resolution  of 
motion  which  are  demonstrated  in  Chap.  HI,  p.  44,  admit  of 
being  satisfactorily  confirmed  by  experiment.  Let  two  small 
wheels  M,  N.  (Fig.  134,)  be  attached  to  a  wall  or  board.  Let  a 
thread  be  passed  over  them,  having  weights  A  and  B,  hooked 
upon  loops  at  its  extremities.  From 
any  part  P  of  the  thread,  let  a  weight 
C  be  suspended,  in  such  a  manner  as 
to  be  in  equilibrio  with  A  and  B.  The  M 
weight  C,  therefore,  is  the  resultant  of 
the  forces  A  and  B  ;  and  since  its  di- 
rection is  that  of  gravity,  it  will  be  rep- 
resented by  a  line  drawn  directly  up- 
ward from  P.  From  P,  on  the  line  A 
PO,  take  PC,  having  as  many  inches  as 
there  are  ounces  in  C  ;  and  from  c  draw 
ca  parallel  to  NP  and  cb  parallel  to  PM. 

*  In  the  same  manner  two  persons  sitting  in  a  boat  under  sail,  toss  a  ball  from 
one  to  the  other  in  the  same  manner  as  though  they  were  at  rest  on  land. 


192  NATURAL   PHILOSOPHY. 

If  the  sides  Pa  and  Pb  of  the  parallelogram  thus  formed  be  mea- 
sured, it  will  be  found  that  Pa  will  consist  of  as  many  inches  as 
there  are  ounces  in  A,  and  Pb  of  as  many  inches  as  there  are 
ounces  in  B ;  consequently,  the  lines  PC,  Pa,  and  Pb,  have  the 
same  ratio  to  each  other  as  the  forces  or  weights  C,  A,  and  B. 
But  PC  is  the  diagonal,  and  Pa  and  Pb  are  the  sides  of  a  paral- 
lelogram. Hence  the  diagonal  of  a  parallelogram  represents  a 
force  equivalent  to  the  two  forces  represented  by  the  sides.* 

This  experiment  is  an  illustration  of  the  composition  of  forces 
rather  than  of  the  composition  of  motions  ;  but  a  simple  experi- 
ment will  show  that  the  same  law  holds  good  with  respect  to  a 
body  actually  set  in  motion  by  two  different  forces.  A  ball  is 
placed  at  one  of  the  corners  of  a  smooth  table.  To  the  same 
corner  are  attached  two  springs,  respectively  in  the  line  of  the 
two  sides  of  the  table,  and  capable  of  giving  a  simultaneous  im- 
pulse to  the  ball.  The  springs  moreover  are  so  proportioned  to 
each  other,  that  one  will  drive  the  ball  across  one  side  of  the 
tabfc,  in  the  same  time  that  the  other  will  drive  the  ball  across 
the  other  side.  Now  on  letting  go  both  springs  at  once,  the  ball 
will  pass,  in  the  same  time,  across  the  diagonal  of  the  table  to 
the  opposite  corner. f 

:<Z05.  We  daily  observe  examples  strikingly  illustrative  of  these 
laws.  In  crossing  a  river,  the  boatman  heads  up  the  stream, 
and  so  combines  the  direction  of  the  boat  with  that  of  the  cur- 
rent, as  to  move  directly  across  in  a  line,  which  is  the  diagonal 
between  the  two.  Rowing,  swimming,  and  flying,  are  severally 
instances  of  motion  in  the  diagonal  between  two  forces.  In  feats 
of  horsemanship,  when  the  rider  leaps  up  from  his  saddle,  we 
are  surprised  not  to  see  the  horse  pass  from  under  him  ;  but  he 
retains  the  motion  he  has  in  common  with  the  horse,  and  does 
not  in  fact  ascend  perpendicularly,  but  obliquely,  rising  in  one 
diagonal,  and  falling  in  another.  In  the  common  i'eats  of  jump- 
ing through  a  hoop,  and  alighting  again  on  the  saddle,  an  inex- 
perienced rider  would  be  likely  to  project  his  body  forward  in 
the  same  manner  as  he  would  do  in  leaping  through  the  same 
hoop  from  the  ground.  In  such  a  ease,  instead  of  alighting  on 
the  saddle,  he  would  alight  either  before  the  horse  or  upon  his 
head  or  neck.  All  that  is  requisite  in  order  to  execute  this  feat, 
is  to  leap  directly  upward  from  the  saddle  to  a  sufficient  height 
to  clear  the  lower  part  of  the  hoop  with  the  feet.  By  the  speed 
which  the  rider  has  in  common  with  the  horse,  his  body  will, 
without  any  exertion  on  his  part,  pass  through  the  hoop,  and  he 
will  alight  again  on  the  saddle,  on  the  other  side;  in  his  de- 
scent.:}; 

266.  The  sailing  of  a  ship  affords  an  instructive  illustration  of 
*  Lardner's  Mech.  p.  51        t  Lib.  Useful  Knowledge,  Mechanics,  p,  5.      t  Ib. 


MECHANICS. 


193 


the  principles  of  the  composition  and  resolution  of  motion. 
When  a  ship  sails  in  the  same  direction  as  the  wind,  she  is  said 
to  be  scudding,  or  sailing  before  the  wind,  and  if  she  had  but  one 
sail,  it  would  act  with  the  greatest  advantage,  when  perpendic- 
ular or  nearly  so  to  the  wind.  When  a  ship  advances  against 
the  wind,  and  endeavors  to  proceed  in  the  nearest  direction  pos- 
sible to  the  point  of  compass  from  which  the  wind  blows,  she  is 
said  to  be  close  hauled.  A  large  ship  will  sail  against  the  wind 
with  her  keel  at  an  angle  of  six  points  with  the  direction  of  the 
wind,  and  sloops  and  smaller  vessels  may  sail  much  nearer. 
When  a  ship  is  neither  sailing  before  the  wind,  nor  is  close 
hauled,  she  is  said  to  be  sailing  large.  In  this  case,  her  sails  are 
set  in  an  oblique  position,  between  the  direction  of  the  wind  and 
that  of  the  intended  course  ;  as  represented  in  the  various  plans 
of  vessels  in  Fig.  135,  where  the  direction  of  the  wind  is  repre- 
sented by  the  arrow,  and  the  position  of  the  yards  and  sails, 
which  are  necessary  for  proceeding  on  the  various  points  of 
compass,  are  shown  by  the  transverse  line  on  each  plan.  Tlie 
relation  of  the  wind  to  the  course  of  the  vessel,  is  determined 
by  the  number  of  points  of  the  compass  between  the  course  she 
is  steering,  and  the  course  she  would  be  steering  if  close  hauled. 
Fig.  135. 


In  Fig.  135,  the  ships  a  and  b  are  close  hauled,  and  the  ships  c 
and  d  (the  former  steering  east  by  north,  and  the  latter  west  by 

25 


194 


NATURAL   PHILOSOPHY. 


north)  have  the  wind  one  point  large.  The  ships  e  and  f,  one 
steering  east  and  the  other  west,  have  the  wind  two  points  large. 
In  this  case,  the  wind  is  at  right  angles  with  the  keel,  and  is  said 
to  be  upon  the  beam.  The  "ships  g  and  h,  steering  southeast  and 
southwest,  have  the  wind  six  points  large,  or,  as  it  is  commonly 
termed,  upon  the  quarter,  and  this  is  considered  as  a  very  favora- 
ble manner  of  sailing,  because  all  the  sails  co-operate  to  increase 
the  ship's  velocity,  whereas  when  the  wind  is  directly  aft,  as  in 
the  vessel  m,  it  is  partly  intercepted  by  the  after  sails,  and  pre- 
vented from  striking  with  its  full  force  on  those  which  are  for- 
ward.* 

267.  To  one  who  has  never  studied  the  doctrine  of  the  com- 
position and  resolution  of  forces,  it  is  apt  to  appear  mysterious 
that  a  ship  is  able  to  sail  with  a  wind  partly  ahead,  and  still 
more  that  two  ships  are  able  to  sail  in  exactly  opposite  direc- 
tions by  the  same  wind.  It  is  proposed  to  explain  these  pheno- 
mena. Let  AB  (Fig.  136)  represent  the  keel  of  a  ship,  and  CD 
the  sail ;  and  let  the  wind  come  in  from  the  side,  in  the  direc- 
tion of  HD.  Let  DE  represent  the  whole  force  of  the  wind, 
and  resolve  DE  into  two  forces,  viz.  into  EF  perpendicular,  and 
FD  parallel  to  the  sail  DC.  Then  it  is  manifest  that  EF  alone 
represents  the  effective  force  of  the  wind  upon  the  sail.  But 
EF  is  not  wholly  employed  in  urging  the  ship  forward,  since  it 
is  oblique  to  her  course  ;  therefore,  again  resolve  EF  into  FG 
parallel  with  the  course  and  GE  at  right  angles  with  it.  The 
latter  force  is  lost  by  the  lateral  resistance  of  the  water,  while 
.FG  is  employed  in  propelling  the  ship  on  her  way. 

Fig.  136. 
V 


By  inspecting  Fig.  136,  it  will  readily  be  seen  that  another 
ship  may  sail  in  the  opposite  direction  by  the  same  wind.  When 

*  Bigelow's  Elements  of  Technology,  p  218. 


MECHANICS.  195 

the  wind  strikes  the  sail  at  right  angles,  or  in  the  direction  EF 
then  only  one  resoiution  is  necessary  ;  for  if  EF  represents  the 
whole  force  of  the  wind,  FG  will  represent  the  force  that  propels 
the  ship  forward,  while  GE  will  represent  the  part  which  is  lost 
by  the  lateral  resistance  of  the  water. 

Since,  resolving  the  force  of  the  wind  after  the  foregoing  man- 
ner, the  effective  part  of  the  force,  viz.  FG,  will  not  wholly  dis- 
appear until  the  wind  is  directly  ahead,  it  might  seem  possible 
to  sail  much  nearer  the  wind  than  is  found  to  be  actually  practi- 
cable. But,  though,  on  account  of  the  peculiar  shape  of  vessels, 
the  forward  resistance  is  much  less  than  the  lateral,  yet  it  is 
something,  and  therefore  requires  more  or  less  of  the  force  that 
acts  parallel  with  the  keel  to  overcome  it. 

268.  The  doctrine  of  the  composition  and  resolution  of  motion, 
by  reducing  a  great  number  of  complicated  motions  to  one,  or 
by  enabling  us  to  estimate  the  precise  influence  of  forces  that 
act  obliquely,  has  greatly  simplified  inquiries  in  Mechanics,  and 
proportionally  advanced  the  science.     It  is  only  by  such  means 
that  the  complex  motions  of  the  heavenly  bodies,  and  the  equally 
diverse  forces  that  control  them,  could  ever  have  been  understood. 
The  subject  has  also  an  extensive  and  important  application  in 
estimating  the  powers  of  machinery.     Since,  for  example,  any 
one  side  of  a  triangle  or  polygon,  is  always  less  than  the  sum  of 
all  the  remaining  sides,  it  follows,  other  things  being  equal,  that 
a  mechanical  effect  will  always  be  more  economically  produced 
by  a  single  force  acting  in  the  proper  direction,  than  by  a  number 
of  forces  acting  in  different  directions. 

269.  By  Art.  55,  it  appears,  that  a  body  acted  upon  at  the  same 
time  by  three  forces,  represented  in  quantity  and  direction  by  the 
three  sides  of  a  triangle  taken  in  order,  will  remain  at  rest.     A  kite 
at  rest  in  the  air,  is  commonly  mentioned  as  an  example  of  this, 
the  three  forces  being,  the  direction  of  the  wind,  the  weight  of 
the  kite,  and  the  action  of  the  string.     Let  AB,  (Fig.  137,)  be  a 
kite,  held  by  the  string  AS.     Let  DF  represent  the  force  of  the 
wind  blowing  horizontally,  and  resolve  it  into  two  forces,  viz. 


196  NATURAL   PHILOSOPHY. 

DC  perpendicular,  and  CF  parallel  to  the  kite.  Then  DC  will 
be  the  only  effective  part  of  the  wind,  since  that  part  which  acts 
parallel  to  the  kite,  can  have  no  influence  on  its  motions.  Again, 
resolve  CD  into  two  forces,  namely,  CE  perpendicular,  and  DE 
parallel  to  the  horizon.  Then  CE  will  represent  the  upward 
force  of  the  wind,  and  DE  its  force  in  a  horizontal  direction. 
Now  when  the  string  AS  makes  such  an  angle  with  the  kite  that 
its  downward  force  AG,  added  to  the  weight  of  the  kite,  shall 
equal  CE,  and  its  horizontal  force  HG,  equal  DE,  the  kite  will 
be  at  rest. 

270.  When  two  motions  which  are  not  in  the  same  straight 
line,  are  combined,  one  of  which  is  uniform,  and  the  other  accel- 
erated, the  moving  body  describes  a  curve.  (Art.  48.)  If  two 
accelerating  forces  act,  in  which  the  rate  of  acceleration  is  the 
same,  the  motion  is  rectilinear ;  but  if  the  rate  of  acceleration  is 
different,  the  motion  is  still  curvilinear.  The  nature  of  the  curve 
will  depend  upon  the  proportion  between  the  uniform  and  ac- 
celerating forces. 


CHAPTER  m. 

OF  THE  CENTER  OF  GRAVITY. 

. 

x571.  THE  principles  which  have  been  discovered  respecting 
the  composition  and  resolution  of  forces,  and  respecting  the 
center  of  gravity,  have  alike  contributed  greatly  to  simplify  the 
doctrines  of  Mechanics.  It  is  characteristic  of  a  great  and  pen- 
etrating mind,  to  devise  means  of  divesting  intricate  subjects  of 
their  complexity,  and  thus  to  bring  easily  within  the  grasp  of 
the  mind,  subjects  otherwise  too  much  involved  to  be  within  its 
comprehension.  Ry  the  rule  of  simple  multiplication,  we  easily 
multiply  any  number  by  one  thousand :  indeed,  it  is  nothing 
more  than  to  annex  three  ciphers  to  the  number  itself;  but  how 
tedious  would  be  this  process,  were  the  rule  of  multiplication 
undiscovered,  and  we  were  unacquainted  with  any  other  method 
of  arriving  at  the  result,  except  to  add  the  given  number  to  it- 
self one  thousand  times.  In  like  manner,  by  means  of  the  rules 
for  the  composition  of  motion,  we  are  enabled  to  reduce  a  thou- 
sand different  motions  to  one ;  and  by  the  doctrine  of  the  center 
of  gravity,  we  are  taught  how  we  may  make  a  force,  situated 
at  one  single  point,  equivalent  to  an  infinite  number  of  forces, 


MECHANICS.  197 

situated  in  as  many  different  points.  And,  instead  of  pursuing 
the  endless  diversities  of  motions  to  which  the  different  parts  of 
a  complicated  system  of  bodies  may  be  subject,  we  are  taught 
how  to  follow  merely  the  motions  of  a  single  individual  point. 
By  the  earth's  attraction,  all  the  particles  which  compose  the 
mass  of  a  body,  are  solicited  by  equal  forces  in  parallel  directions 
downward.  If  these  component  particles  were  placed  in  mere 
juxtaposition,  without  any  mechanical  connection,  the  force  im- 
pressed on  any  one  of  them,  could  in  no  wise  affect  the  others, 
and  the  mass  would,  in  such  a  case,  be  contemplated  as  an  ag- 
gregation of  small  particles  of  matter,  each  urged  by  an  inde- 
pendent force.  Then,  according  to  Art.  60,  the  resultant  consti- 
tutes another  force  parallel  to  the  others.  But  the  bodies  which 
are  the  subjects  of  investigation  in  mechanical  science,  are  not 
found  in  this  state.  Solid  bodies  are  coherent  masses,  the  par- 
ticles of  which  are  firmly  bound  together,  so  that  any  force  which 
affects  one,  being  modified  according  to  circumstances,  will  be 
transmitted  through  the  whole  body.* 

272.  As  all  bodies  which  are  subjects  of  mechanical  inquiry, 
on  the  surface  of  the  earth,  must  be  continually  influenced  by 
terrestrial  gravity,  it  is  desirable  to  obtain  some  easy  and  sum- 
mary method  of  estimating  the  effects  of  this  force.     To  consider 
it,  as  is  unavoidable  in  the  first  instance,  the  combined  action  of 
an  infinite  number  of  equal  and  parallel  forces,  soliciting  the  ele- 
mentary molecules  downward,  would  be  attended  with  manifest 
inconvenience.     An  infinite  number  of  forces,  and  an  infinite 
subdivision  of  the  mass,  would  form  parts  of  every  mechanical 
problem.     To  overcome  this  difficulty,  and  to  obtain  all  the  ease 
and  simplicity  which  can  be  desired  in  elementary  investigations, 
it  is  only  necessary  to  determine  some  force,  whose  single  effect 
shall  be  equivalent  to  the  combined  effect  of  the  gravitation  of 
all  the  molecules  of  the  body.f     Such  a  force  is  obtained  by  sup- 
posing all  the  action  to  be  concentrated  in  the  center  of  gravity. 
We  have  already  defined  it  thus :  the  center  of  gravity  of  a  body, 
is  that  point  about  which,  if  supported,  all  the  parts  of  a  body  (acted 
upon  only  by  the  force  of  gravity)  would  balance  each  other  in  any 
position.^     (Art.  64.) 

273.  To  find  the  center  of  gravity  by  experiment,  several  differ- 

*  Lardner's  Elements  of  Mechanics,  p.  107.  t  Ib.  p.  108. 

\  Others  define  it  to  be  "  the  point  through  which  passes  the  resultant  of  all  the  par- 
ticular forces  exerted  by  the  gravity  of  the  several  parts  of  the  body,  or  system  of 
bodies,  in  whatever  position  the  body  or  system  is  placed."  (Cambridge  Mech.  p.  45.) 

"  The  resultant  of  any  number  of  parallel  forces  continues  of  the  same  intensity, 
and  passes  through  the  same  point,  whatever  be  the  direction  of  the  forces  ;  hence  it 
is.  called  the  Center  of  Parallel  Forces.  When  the  body  is  moving  forward  in  a 
straight  line,  under  the  action  of  forces  other  than  gravity,  it  is  called  the  center  of 
Inertia ;  when  the  body  is  acted  upon  by  gravity,  it  is  called  the  center  of  gravity." 
(Renwick  on  the  Steam  Engine,  p.  14.) 


198  NATURAL   PHILOSOPHY. 

ent  methods  present  themselves.  We  will  first  suppose  the  body 
to  be  in  the  shape  of  a  piece  of  board  of  uniform  thickness.  Sus- 
pend it  by  one  corner,  and  from  the  same  corner  let  fall  a  plumb 
line,  and  mark  its  line  of  direction  on  the  surface  of  the  board. 
Suspend  the  board  from  any  other  point,  and  mark  the  line  of 
direction  of  the  plumb  line  as  before,  and  the  point  where  these 
lines  intersect  each  other,  must  obviously  be  the  center  of  gravity, 
since  that  center  is  in  both  of  the  lines.  (Art.  65.) 

But  when  the  body  is  not  of  uniform  thickness,  but  is  any 
irregular  solid,  suspend  the  body  by  a  thread,  and  let  a  small 
hole  be  bored  through  it,  in  the  exact  direction  of  the  thread,  so 
that  if  the  thread  were  continued  below  the  point  where  it  is 
attached  to  the  body,  it  would  pass  through  this  hole.  The  body 
being  successively  suspended  by  several  different  points  in  its 
surface,  let  as  many  small  holes  be  bored  through  it  in  the  same 
manner.  If  the  body  be  then  cut  through,  so  as  to  discover  the 
directions  which  the  several  holes  have  taken,  they  will  all  be 
found  to  cross  each  other  at  one  point  within  the  body.  Or  the 
same  fact  may  be  discovered  thus  :  a  thin  wire  which  nearly 
fills  the  holes  being  passed  through  any  one  of  them,  will  be 
found  to  intercept  the  passage  of  a  similar  wire  through  any 
other.* 

A  convenient  method  of  finding  the  center  of  gravity  of  a  bo^y 
is  to  balance  it  in  different  positions  across  a  thin  edge,  as  the 
edge  of  a  knife  or  a  prism.  The  same  thing  may  be  effected, 
when  the  shape  of  the  body  will  admit  of  it,  by  laying  it  on  the 
edge  of  a  table,  and  letting  so  much  of  it  project  over  the  edge, 
that  the  slightest  disturbance  will  cause  it  to  fall.  The  center  of 
gravity  is  the  point  in  which  the  several  lines  marked  on  the 
body,  where  the  edge  cuts  it,  intersect  one  another.  From  some 
or  all  of  the  foregoing  trials,  the  center  of  gravity  of  bodies  may 
be  nearly  ascertained ;  but  in  order  to  find  it  with  absolute  exact- 
ness, we  are  frequently  obliged  to  resort  to  intricate  mathematical 
processes. 

274.  By  whatever  method  the  center  of  gravity  of  a  body  has 
been  ascertained,  we  shall  find  that  when  that  is  supported,  the 
body  will  remain  at  rest  in  every  position.  Thus  a  globe  will 
stand  securely  on  a  very  small  perpendicular  support,  since  that 
support  will  necessarily  be  under  the  center  of  gravity ;  a  lever, 
as  the  beam  of  a  balance,  poised  on  its  center  of  gravity,  will  be 
at  rest  in  every  position  it  takes  while  turning  round  the  fulcrum, 
and  however  irregular  the  body  may  be,  it  will,  when  balanced 
on  its  center  of  gravity,  obstinately  maintain  its  position. 

When  a  body  is  suspended  by  an  inflexible  rod  from  a  center 
of  motion,  and  revolves  around  it,  it  will  be  at  rest  only  when 

*  Lardner's  Mech.  p.  110. 


MECHANICS.  199 

the  center  of  gravity  is  either  directly  below,  or  directly  above  the 
center  of  motion.  For  it  is  only  in  these  two  cases,  that  the  cen- 
ter of  gravity  will  be  in  the  line  which  is  drawn  through  the 
center  of  motion  perpendicular  to  the  horizon.  The  stationary 
point  above  the  center  of  motion  is  very  unstable,  since  the 
slightest  disturbing  force  throws  the  body  out  of  the  line  of  di- 
rection, when,  by  the  force  of  gravity,  it  immediately  descends 
to  the  lowest  point  it  can  reach,  and  vibrates  about  that  point 
until  it  finally  settles  itself  with  the  centre  of  gravity  immediately 
under  the  point  of  suspension  ;  and  whenever  it  is  thrown  out  of 
this  position  the  same  vibrations  are  renewed  until  it  resumes  it. 
When  therefore  the  center  of  gravity  is  at  the  lowest  point  it  is 
capable  of  reaching,  the  equilibrium  is  stable,  since  the  body  ob- 
stinately maintains  that  position.  On  this  principle,  gates  which 
have  their  center  of  gravity  raised  as  they  are  opened,  shut  spon- 
taneously. 

275.  The  stability  of  a  body^  not  only  requires  that  the  center 
of  gravity  should  be  low,  but  that  the  line  of  direction  (or,  the 
line  which  is  drawn  through  the  center  of  gravity  perpendicular 
to  the  horizon)  should  fall  within  the  base.  (Art.  69.)     The  far- 
ther it  falls  from  the  extremity  of  the  base,  the  more  stable  is  the 
position.     Hence  the  stability  of  a  pyramid,  when  standing  on 
its  broad  base,  and  its  instability  when  inverted.     For  the  same 
reason,  all  broad  vessels,  as  steamboats,  are  difficult  to  upset, 
while  vehicles  with  narrow  bases  are  easily  overturned.     When 
a  load  is  so  situated  as  to  raise  the  center  of  gravity,  it  increases 
the  liability  to  upset,  because  it  increases  the  facility  with  which 
the  line  of  direction  is  thrown  without  the  base.     Thus  carts 
loaded  with  hay,  or  bales  of  cotton,  are  very  liable  to  be  over- 
turned.    The  same  is  true  of  stages  carrying  passengers  or  bag- 
gage on  the  top.     On  the  other  hand,  a  large  ship  well  supplied 
with  ballast,  is  capsized  with  great  difficulty,  since  the  center  of 
gravity  of  all  parts  of  the  ship  is  so  low,  as  to  render  it  difficult 
to  throw  the  line  of  direction  without  the  base.     Yet  if  the  cen- 
ter of  gravity  is  very  low,  a  ship  will  rock  excessively  in  a  rough 
sea,  since  the  upper  parts  near  the  deck,  move  over  a  greater 
space  in  proportion  as  their  distance  from  the  centre  of  gravity 
is  greater. 

276.  There  are  many  remarkable  structures  which  lean  or  in- 
cline a  little  ;  but  so  long  as  the  line  of  direction  falls  within  the 
base,  and  the  parts  of  the  mass  have  sufficient  tenacity  among 
themselves  to  hold  together,  the  structure  will  stand.     The  fa- 
mous tower  of  Pisa  was  built  intentionally  inclining,  to  frighten 
and  surprise :  with  a  height  of  one  hundred  and  thirty  feet,  it 
overhangs  its  base  sixteen  feet.*     This  circumstance  greatly  en- 

*  Some  travellers,  however,  are  of  opinion  that  the  inclination  of  the  tower  of  Pisa, 


200  NATURAL   PHILOSOPHY. 

hances  the  emotion  of  the  spectator  from  its  summit.  Many  an- 
cient spires  and  other  tall  structures,  are  found  to  have  lost  some- 
thing of  their  perpendicularity. 

Rocking  stones  are  rocks  which  are  sometimes  found  so  exactly 
poised  upon  their  center  of  gravity,  that  a  very  small  force  is 
sufficient  to  put  them  in  motion.*  The  rocking  of  a  balloon 
when  it  begins  to  ascend,  affords  an  illustration  of  the  tendency 
of  bodies  to  vibrate  around  the  center  of  gravity. 

277.  The  motions  of  animals  are  regulated  in  conformity  with 
the  doctrines  of  the  center  of  gravity.     A  body  is  seen  tottering 
in  proportion  as  it  has  great  altitude  and  a  narrow  base  ;  but  it  is 
a  peculiarity  in  man  to  be  able  to  support  his  figure  with  great 
firmness  on  a  very  narrow  base,  and  under  constant  changes  of 
attitude.     This  faculty  is  acquired  slowly,  because  of  the  diffi- 
culty.    The  great  facility  with  which  the  young  of  quadrupeds 
walk,  is  ascribed  in  part  to  their  broad  supporting  base.     Many 
of  our  most  common  motions  and  attitudes,  depend  for  their  ease 
and  gracefulness  upon  a  proper  adjustment  of  the  center  of  grav- 
ity.    The  erect  posture  of  a  man  carrying  a  load  upon  his  head — 
leaning  to  one  side  when  a  heavy  weight  is  carried  in  the  oppo- 
site hand — leaning  forward  when  a  weight  is  on  the  back — or 
backward  when  the  weight  is  in  the  arms  ;  these  are  severally 
examples  in  point.     When  a  man  rises  from  his  chair,  he  brings 
one  foot  back,  and  leans  the  body  forward,  in  order  to  bring  the 
center  of  gravity  over  the  base ;  and  without  adjusting  it  in  this 
manner,  it  is  hardly  possible  to  rise.     A  man  standing  with  his 
heels  close  to  a  perpendicular  wall,  cannot  bend  forward  suffi- 
ciently to  pick  up  any  object  that  lies  on  the  ground  near  him 
without  himself  falling  forward. 

The  art  of  rope  or  wire  dancing,  depends  in  a  great  degree  upon 
a  skilful  adjustment  of  the  centre  of  gravity.  The  rope  dancer 
frequently  carries  in  his  hand  a  stick  loaded  with  lead,  which  he 
so  manages  as  to  counterbalance  the  inclinations  of  his  body 
which  would  throw  the  line  of  direction  out  of  the  base.  Upon 
a  similar  principle,  the  equestrian  balances  himself  on  one  foot 
on  a  galloping  horse. 

The  vegetable  creation  is  subject  also  to  these  general  laws  of 
nature.  Trees,  by  the  weight  and  height  of  their  tops,  would 
seem  peculiarly  liable  to  fall ;  but  their  roots  afford  a  correspond- 
ing breadth  of  base,  while  their  perpendicular  trunks,  and  the 
symmetrical  disposition  of  the  branches,  conspire  to  increase  their 
stability. 

278.  The  position  of  the  center  of  gravity  of  any  number  of 

is  the  effect  of  time.     It  is  said  that  an  ancient  picture  of  the  tower  represents  it  aa 
erect.     Arnott's  El.  Phys.  p.  121. 
*  Amer.  Journal  of  Science,  Vol.  7 


MECHANICS.  201 

separate  bodies,  is  never  altered  by  the  mutual  action  of  those 
bodies  on  each  other.  If,  for  example,  two  bodies,  by  mutual 
attraction,  approach  each  other,  the  center  of  gravity  remains  at 
rest,  until  finally  the  bodies  meet  in  this  point.  If,  by  their  mu- 
tual action,  they  contribute  to  make  each  other  revolve  in  orbits, 
it  is  around  their  common  center  of  gravity.  Thus  the  earth 
and  moon  revolve  around  a  common  .center  of  gravity :  the  same 
is  true  of  the  sun  and  all  the  bodies  that  compose  the  solar  sys- 
tem. Were  the  centrifugal  force  to  be  suspended,  and  the  bodies 
abandoned  to  the  mutual  action  of  each  other,  they  would  all 
meet  in  their  common  center  of  gravity.  (Art.  81.)  This  nat- 
urally results  from  the  principle  that  the  momenta  on  opposite 
sides  of  the  center  of  gravity  are  equal,  and  that  bodies  by  their 
mutual  action  produce  equal  momenta  in  each  other. 

279.  The  doctriwes  of  the  center  of  gravity  suggest  the  readi- 
est method  of  solving  a  great  number  of  PRACTICAL  PROBLEMS. 

Suppose  three  persons  were  carrying  a  stick  of  timber,  (A  by 
himself  supporting  one  end,  and  B  and  C  by  a  handspike  lifting 
together  toward  the  other  end,)  and  it  were  required  to  deter- 
mine at  what  distance  from  the  end  of  the  stick  the  handspike 
must  be  placed  in  order  that  the  three  persons  might  bear  equal- 
ly. A  stick  of  timber  being  a  body  of  regular  shape  and  uni- 
form density,  has  its  center  of  gravity  coincident  with  the  center 
of  magnitude.  We  may  therefore  proceed  on  the  supposition 
that  the  entire  weight  is  collected  in  the  center.  Now  in  order 
that  B  and  C  may  together  lift  twice  as  much  as  A,  they  must 
be  twice  as  near  the  center.  But  the  distance  of  A  from  the 
center  is  one  half  the  length  of  the  stick  ;  therefore  the  distance 
of  the  required  point  from  the  center  is  one  fourth  the  length  of 
the  stick,  and  consequently  it  is  one  fourth  the  same  length  from 
the  end  of  the  stick. 

The  result  thus  obtained  from  theory,  may  be  easily  submitted 
to  the  test  of  experiment.  For  if  we  take  the  weight  of  the  stick 
of  timber  with  a  pair  of  steelyards,  and  then,  resting  one  end  on 
some  support,  attach  a  cord  at  the  distance  of  one  fourth  of  the 
length  of  the  stick  from  the  other  end,  and  thus  connect  the  stick 
with  the  steelyards,  we  shall  find  the  weight  equal  to  two  thirds 
of  the  whole.* 

280.  The  method  of  finding  the  areas  of  the  surfaces  and  the 
solid  contents  of  bodies,  particularly  of  such  as  are  of  unusual 
figure,  may  frequently  be  very  much  simplified,  by  applying  the 

*  This  experiment  may  be  repeated  with  much  precision  in  the  following  manner. 
Take  a  Gunter's  scale,  well  made,  and  ascertain  its  weight  by  a  delicate  balance. 
Let  one  end  rest  on  a  sharp  edge,  as  the  edge  of  a  prism,  and  attaching  a  string  at 
the  distance  of  one  fourth  the  length  of  the  scale  from  the  other  end,  connect  it  with 
one  of  the  arms  of  the  balance  :  the  weight  will  be  exactly  two  thirds  of  the  whole. 

26 


202 


NATURAL   PHILOSOPHY. 


principles  of  the  center  of  gravity.  This  is  sometimes  called 
the  centrobaryc  method.  It  was  discovered  by  Pappus,  an  an- 
cient mathematician  of  Alexandria,  but  was  more  completely 
iiscussed  and  illustrated  by  Guldinus,  Professor  of  Mathematics 
at  Rome,  about  the  year  1640.*  This  remarkable  property  of 
the  center  of  gravity  is  expressed  in  the  following  propositions : 

1.  If  any  line  whatsoever  revolves  about  a  fixed  point,  the  SURFACE 
which  it  generates  is  equal  to  the  product  of  the  given  line  into  the 
circumference  described  by  its  center  of  gravity. 

2.  If  a  surface  having  any  figure  whatsoever,  revolves  about  an 
axis,  the  SOLID  generated,  is  equal  to  the  product  of  that  surface  into 
the  circumference  described  by  its  center  of  gravity. 

Thus,   the    straight   line   CD,  Fi    138 

(Fig.  138,)  revolving  about  tlpe      ...  A 

center  C,  describes  a  circle  whose 
surface  is  equal  to  CD  into  the 
circumference  of  the  circle  de- 
scribed by  its  center  of  gravity  E. 
This  is  evident  also  from  the  con- 
sideration that,  since  E  is  the 
center  of  the  line  CD,  the  cir- 1 


cumference  described  by  it  will 
be  half  the  length  of  the  circum- 
ference ADB  ;  and  the  area  of  a 
circle  is  equal  to  the  product  of 
the  radius  into  half  the  circum- 
ference. 

Again,  the  small  circle,  having 
its  center  coincident  with  the  extremity  of  the  line  D,  and  re- 
volving round  on  the  circumference  of  the  circle  described  by 
CD,  being  everywhere  perpendicular  to  the  plane  of  the  circle, 
would  describe  a  solid  figure  like  the  ring  of  an  anchor  ;  and  the 
line  described  by  the  center  of  gravity,  that  is,  the  circumference 
of  the  circle,  multiplied  into  the  area  of  the  revolving  figure, 
would  give  the  solidity  of  the  ring.  In  like  manner,  in  a  cone, 
the  solidity  is  equal  to  the  area  of  the  generating  triangle,  mul- 
tiplied into  the  circumference  of  the  circle,  formed  by  the  revo- 
lution of  the  center  of  gravity  of  the  triangle  ;  and  the  surface- 
is  equal  to  the  product  of  the  perimeter  of  the  same  triangle, 
multiplied  by  the  circumference  described  by  the  center  of  grav- 
ity of  the  same  perimeter. 

*  Hence  these  properties  of  the  center  of  gravity  are  sometimes  called  Guldinus'e 
Properties 


MECHANICS.  203 


CHAPTER  IV. 

OF    MACHINERY. 

281.  THE  organs  employed  in  communicating  motion,  are  tools, 
machines,  and  engines.  .  Tools  are  the  simplest  instruments  of 
art ;   these,  when  complicated  in  their  structure,  become  ma- 
chines ;  and  machines,  when  they  act  with  great  power,  take  the 
name  of  engines.     Among  the  ancients,  machines  were  confined 
chiefly  to  the  purposes  of  architecture  and  war ;  and  they  were 
moved  almost  exclusively  by  the  strength  of  animals.     Thus,  in 
building  one  of  the  great  Pyramids  of  Egypt,  vast  masses  of 
stone  were  raised  to  a  great  height,  amounting  together    to 
10,400,000  tons.    In  this  labor  were  employed  100,000  men  for  20 
years.     The  advantage  which  man  has  gained  by  pressing  into 
his  service  the  great  powers  of  nature,  instead  of  depending  on 
his  own  feeble  arm,  is  evinced  by  the  fact,  that  by  the  aid  of  the 
steam  engine,  one  man  can  now  accomplish  as  much  labor  as 
27,000  Egyptians,  working  at  the  rate  at  which  they  built  the 
pyramids.*     In  war  also,  while  the  use  of  gunpowder  was  un- 
known, engines  of  great  power  were  invented  for  throwing 
stones  and  javelins,  and  for  demolishing  fortifications.     Such 
were  the  Catapulta,  the  Balista,  and  the  Battering  Ram,  of  the 
Romans.     Yet  it  is  remarkable,  that  during  many  ages,  while 
such  powerful  auxiliaries  were  employed  in  architecture  and  in 
war,  the  ancients  should  have  made  so  little  use  as  they  did  of 
machinery  in  the  ordinary  processes  of  the  arts.     Nor  did  the 
philosophers  of  antiquity  cultivate  science  with  any  reference  to 
the  improvement  of  the  arts,  an  object  which  they  considered 
below  the  dignity  of  true  philosophy ;  but  Lord  Bacon  was  the 
first  to  show  that  "  a  principle  in  science  is  a  rule  in  art." 

282.  The  Mechanical  Powers,  being  the  principal  instruments 
of  art,  will  first  require  our  attention.     They  have  already  been 
considered  theoretically  ;  we  are  now  to  consider  them  practical- 
ly.    It  will  be  recollected  that,  when  two  forces  act  on  one  an- 
other by  means  of  any  machine,  that  which  gives  motion  is  called 
the  power ;  that  which  receives  motion,  the  weight.     (Art.  97.) 
The  weight  includes  not  only  the  proper  weight  of  the  body  or 
bodies  moved,  but  also  every  kind  of  resistance  opposed  to  the 
action  of  the  power,  whether  it  arises  from  the  quantity  of  mat- 
ter in  the  body  moved,  or  from  the  inertia  of  the  machine  itself, 
or  from  the  air,  or  from  friction.     It  will  be  further  recollected, 
that  an  equilibrium  is  produced  between  two  forces,  when  their 
momenta  are  equal.     (Art.  149.)     Now,  since  momentum  is  com- 
pounded of  quantity  of  matter  and  velocity,  a  given  momentum 

*  Dupin. 


204 


NATURAL   PHILOSOPHY. 


may  be  produced,  either  by  giving  a  great  velocity  to  a  small 
weight,  or  a  small  velocity  to  a  great  weight.  The  considera- 
tion of  this  subject  will  be  resumed  hereafter. 


THE    LEVER. 


283.  The  principle  of  the  lever  has  a  most  extensive  applica- 
tion in  the  arts,  and  the  forms  under  which  it  occurs  are  very 
various.  We  may  contemplate  it  as  having  equal  or  unequal 
arms.  The  balance  affords  the  most  common  example  of  a  lever 
with  equal  arms.  The  necessity  of  arriving  at  the  weight  of 
bodies  with  the  greatest  degree  of  accuracy  in  pecuniary  trans- 
actions, and  more  especially  in  delicate  scientific  researches,  as 
those  of  chemical  analysis,  has  induced  men  of  science,  and  art- 
ists, to  bestow  great  and  united  attention  upon  the  construction 
of  this  instrument,  until  they  have  brought  it  to  an  astonishing 
degree  of  perfection. 

Fig.  139. 


284.  The  principal  parts  of  the  balance  are  the  beam  GH, 
(Fig.  139,)  the  points  of  suspension  G  and  H,  and  the  fulcrum  F. 
In  order  to  construct  a  perfect  balance,  the  most  important  par- 
ticulars to  be  attended  to,  are  the  length  of  the  arms,  that  is,  of 
the  beam ;  the  situation  of  the  center  of  gravity  of  the  whole 
instrument,  with  respect  to  the  fulcrum  or  center  of  motion  ;  and 
the  position  of  the  points  of  suspension. 

(1.)  The  sensibility  of  the  balance  is  increased  by  increasing 
the  lengths  of  the  arms ;  but  unless  the  arms,  when  long,  are  at 
the  same  time  of  considerable  weight,  they  will  not  have  the 
requisite  strength,  but  will  be  liable  to  bend  ;  and  an  increase  of 
weight,  adds  to  the  amount  of  friction  on  the  center  of  motion 


MECHANICS.  205 

It  is  not  common,  therefore,  to  make  the  arms  of  a  very  delicate 
balance  more  than  nine  inches  in  length  ;  and,  for  the  purpose 
of  uniting  lightness  with  strength,  the  beam  is  composed  of  two 
hollow  cones  placed  base  to  base,  as  in  Fig.  139. 

(2.)  The  center  of  gravity  of  the  instrument  must  be  a  little 
below  the  center  of  motion.  For  if  the  beam  is  balanced  on  its 
center  of  gravity,  it  will  remain  at  rest  in  every  position,  (Art. 
64,)  whereas  it  must  be  at  rest  only  when  in  a  horizontal  posi- 
tion. If  the  center  of  gravity  is  above  the  center  of  motion,  the 
position  is  too  unstable,  (Art.  274,)  and  upon  the  least  disturb- 
ance of  the  equilibrium,  the  beam  will  be  liable  to  upset.  Final- 
ly, if  the  center  of  gravity  is  too  far  below  the  center  of  motion, 
the  equilibrium  will  be  too  stable.  Hence,  in  very  delicate  bal- 
ances, the  center  of  motion  is  placed  a  little  above  the  center  of 
gravity. 

(3.)  The  points  of  suspension  must  be  in  the  same  right  line 
with  the  center  of  motion.  For  since  when  weights  are  added 
to  the  scales,  the  effect  is  the  same  as  though  they  were  concen- 
trated in  the  points  of  suspension,  (Art.  99,)  were  those  points 
above  the  center  of  motion,  the  center  of  gravity  would  be  liable 
to  be  shifted  above  the  center  of  motion,  when  the  beam  would 
upset ;  and  if  the  same  points  were  below  the  center  of  motion, 
the  center  of  gravity  would  be  too  low,  and  the  equilibrium  too 
stable. 

In  order  to  prevent  friction  as  much  as  possible,  the  fulcrum 
is  made  of  hardened  steel,  and  shaped  into  a  triangular  prism, 
or  knife  edge,  smoothly  rounded,  and  turning  on  a  plane  of  agate 
or  steel,  or  some  other  very  hard  and  polished  substance. 


285.  It  is  only  by  a  nice  attention  to  all  these  particulars,  that 
artists  have  been  able  to  give  to  the  balance  so  great  a  sensibility. 
Some  have  been  made  to  turn  with  the  1000th  part  of  a  grain.*  By 
loading  the  beam,  the  sensibility  of  the  instrument  is  diminished  ; 
it  is  customary,  therefore,  to  estimate  its  power,  by  finding  what 
part  of  the  weight  with  which  it  is  loaded  it  takes  to  turn  it. 
Thus,  if  when  loaded  with  7000  grains,  it  will  turn  with  one 
grain,  its  power  is  T^Vo-'  A  balance  constructed  by  Ramsden,  a 
celebrated  English  artist,  for  the  Royal  Society,  turned  with  the 
ten  millionth  part  of  the  weight.f  Delicate  balances  are  usually 
covered  with  a  glass  case  to  prevent  agitation  from  the  air,  and 
to  secure  them  from  injury.  Fig.  139,  represents  an  instrument 
of  this  kind  made  for  the  Royal  Institution  of  Great  Britain.  J 

*  Nicholson's  Chemical  Diet— Kater  in  Lardner's  Mechanics, 
t  Young's  Lectures  on  Nat.  Phil.  I,  135. 

t  For  a  full  account  of  the  most  accurate  balances,  see  Kater  on  "  Balances  and 
Pendulums"  in  Lardner's  Mechanics,  p.  278. 


206 


NATURAL   PHILOSOPHY. 


286.  The  bent  lever  balance  is  represented  Fig.  140 
in  Fig.  140.     The  weight  C  acts  as  though 

it  were  concentrated  in  the  point  D,  (Art. 
104,)  and  the  weight  in  the  scale  acts  at  K  ;  GJ 
hence  an  equilibrium  will  take  place,  when 
the  article  weighed  has  to  C  the  same  ratio 
as  DB  has  to  BK.  Now  every  increase  of 
weight  added  to  the  scales  causes  C  to  rise 
on  the  arc  FG,  and  D  to  recede  from  B. 
Hence  the  different  positions  of  C,  according 
as  different  weights  are  added  to  the  scale, 
may  be  easily  determined,  and  the  corres- 
ponding numbers  marked  on  the  scale  FG. 

287.  It  is  essential  to  an  accurate  balance,  that  the  two  arms 
should  be  precisely  equal  in  length.     The  false  balance,  which  is 
sometimes  used  with  a  design  to  defraud,  has  its  arms  unequal. 
The  dealer  turns  such  an  instrument  to  his  account,  both  in  buy- 
ing and  selling.     In  buying  he  puts  his  weights  on  the  longer 
side,  for  then  it  takes  more  than  an  equivalent  to  balance  them ; 
and,  in  selling,  he  puts  his  weights  on  the  shorter  side,  because 
less  than  an  equivalent  will  produce  an  equilibrium.     The  fraud 
may  be  detected  by  making  the  weights  and  the  merchandise 
change  places.     The  true  weight  may  be  determined  from  such 
a  balance  by  the  rule  given  on  p.  94 ;  or  more  conveniently,  by 
putting  the  article  whose  weight  is  to  be  determined  into  one 
scale,  and  counterpoising  it  with  sand,  shot,  or  any  convenient 
substance,  in  the  other  scale,  and  then  removing  the  article,  and 
finding  the  exact  weight  of  the  counterpoise.     It  is  evident  that 
the  weight  of  the  merchandise  will  be  the  same  as  that  of  the 
weights  employed  to  balance  its  counterpoise. 

288.  The  steelyard  is  a  lever  having  unequal  arms,  in  which 
the  same  body  is  made  to  indicate  different  weights,  by  placing 
it  at  different  distances  from  the  fulcrum.     A  pair  of  steelyards 
has  usually  two  graduated  sides  for  determining  smaller  or  greater 
weights.     It  will  be  seen  that  on  the  greater  side,  the  weight  is 
placed  nearer  the  fulcrum.     Consequently,  the  weight  indicated 
by  the  counterpoise,  when  at  a  given  distance  from  the  fulcrum, 
will  be  proportionally  greater.     This  instrument  is  very  conve- 
nient, because  it  requires  but  one  weight.     The  pressure  on  the 
fulcrum,  excepting  that  of  the  apparatus  itself,  is  only  that  of  the 
article  weighed,  whereas  in  the  balance,  the  fulcrum  sustains  a 
double  weight.     But  the  balance  is  susceptible  of  more  sensi- 
bility than  the  steelyard,  because  the  subdivisions  of  its  weights 
can  be  effected  with  a  greater  degree  of  precision  than  the  sub- 
division of  the  arms  of  a  steelyard.* 

»  Kater. 


MECHANICS.  207 

289.  The  spring  steelyard  is  a  very  convenient  in-    Fig.  141. 
strument  for  weighing,  in  cases  where  the  subdivisions 

of  the  weights  are  large.  It  depends  on  the  elasticity 
of  a  spiral  steel  spring,  to  compress  or  extend  which 
requires  a  force  proportioned  to  the  degree  of  com- 
pression or  extension.  The  manner  of  applying  it 
will  be  easily  understood  from  the  representation  in 
Fig.  141.  After  continued  use,  especially  when  load- 
ed with  heavy  weights,  the  elasticity  of  the  spring  is 
liable  to  be  impaired,  and  the  accuracy  of  the  instru- 
ment diminished.  When  made,  however,  in  the  best 
manner,  spring  steelyards  retain  their  accuracy  for  a 
long  time. 

290.  In  Fig.  142,  is  represented  a  vertical  section 

of  a  large  Weighing  Machine,  such  as  is  used  for  loads  of  hay, 
cotton,  or  other  heavy  merchandise. 

AB,  a  section  of  the  platform,  resting  loosely  on  a  frame. 

CN,  DN,  levers  of  the  second  kind,  having  their  fulcrums  at 
C,  D,  and  resting  on  a  bar  at  N. 

W,  W,  pins  which  press  upward  against  the  platform  when 
the  levers  are  raised. 

EF,  a  lever  likewise  of  the  second  kind,  having  its  fulcrum  at 
E,  and  connected  by  a  perpendicular  arm,  with  the  beam  of  a 
pair  of  steelyards  at  G. 

Fig.  142. 

ream 


Four  levers  are  usually  employed,  proceeding  from  the  four 
corners  of  an  immovable  frame,  or  having  their  fulcrums  firmly 
set  in  masonry.  The  levers  all  rest  on  the  common  support  at  N. 

Suppose  a  load  of  merchandise  is  placed  on  the  platform  to  be 
weighed.  By  the  steelyards,  we  ascertain  that  the  weight  ex- 
erted at  G  is  100  pounds,  which  force  is  that  exerted  at  F  to 
raise  the  lever  EF.  Supposing,  for  convenience  of  computation, 
that  the  levers  have  their  longer  ten  times  the  length  of  their 
shorter  arms,  then  100  pounds  at  F  balances  a  force  of  1000 


208  NATURAL   PHrLOSOPHY. 

pounds  at  N.  This  force  is  still  further  multiplied  by  the  four 
levers  so  as  to  become  10,000  pounds,  which  is  the  weight  of  the 
load  including  that  of  the  platform.  If  the  platform  rested  on  a 
single  lever,  this  would  of  course  sustain  a  weight  of  10,000 
pounds ;  but  as  the  levers  severally  sustain  the  same  part  of 
the  weight,  each  one  bears  only  one  fourth  of  the  load,  or  2,500 
pounds. 

291.  When  a  weight  is  supported  by  a  lever  which  rests  on 
two  props,  the  pressure  upon  both  fulcrums  is  equal  to  the  whole 
weight.     This  principle  is  sometimes  applied  in  ascertaining  the 
weight  of  a  body  too  heavy  for  the  steelyards.     The  body  is  sus- 
pended immovably  near  the  center  of  a  pole,  and  the  steelyards 
are  applied  to  each  end  of  the  pole  separately,  the  other  end 
meanwhile  resting  on  its  fulcrum.     The  two  weights  being  add- 
ed together,  make  the  entire  weight  of  the  body.     If  the  body 
is  suspended  exactly  in  the  center  of  the  pole,  it  will  be  xufii- 
cient  to  obtain  the  weight  of  one  end  and  double  it.    The  weight 
of  the  lever  should,  in  both  cases,  be  subtracted  from  the  entire 
weight. 

292.  Since  when  a  weight  is  sustained  between  two  props,  the 
part  sustained  by  each  prop  is  inversely  as  the  distance  of  the  weight 
from  it,  it  follows  that  a  load  borne  on  a  pole,  between  two 
bearers,  is  distributed  in  this  ratio.     As  the  effort  of  the  bearers, 
and  the  direction  of  the  weight,  are  always  parallel,  it  makes  no 
difference  whether  the  pole  is  parallel  to  the  horizon  or  inclined 
to  it.    Whether  the  bearers  ascend  or  descend,  or  move  on  a  level 
plane,  the  weight  will  be  shared  between  them  in  the  same  con- 
stant ratio.     (See  Fig.  72.) 

293.  Handspikes  and  crowbars  are  familiar  examples  .of  levers 
of  the  first  kind.     A  hammer  affords  an  example  of  the  bent  lev- 
er ;  and  shears,  pliers,  nutcrackers,  and  all  similar  instruments, 
are  double  levers  ;  that  is,  they  consist  of  two  levers  united.     A 
pair  of  shears,  with  long  handles,  like  those  used  by  tinners,  ex- 
hibit very  strikingly  the  increase  of  power  gained  by  bringing 
the  weight  or  substance  acted  on  nearer  to  the  fulcrum.     The 
jaws  of  animals  exhibit  a  similar  property.     An  oar,  applied  to 
a  boat  rowed  by  hand,  a  wheel-barrow,  and  a  door  shut  by  the 
hand  applied  to  the  edge  remote  from  the  hinges,  severally  fur- 
nish instances  of  levers  of  the  second  kind,  where  the  weight  is 
between  the  fulcrum  and  the  power. 

The  crane  is  a  lever  of  the  second  kind,  which  is  much  used 
when  great  weights  are  transported  for  a  short  distance,  as 
heavy  boxes  of  merchandise  from  a  vessel  to  the  wharf,  or 
great  masses  of  stone  from  the  quarry  to  a  car  or  boat.  An  ex- 
ample  of  the  crane,  on  a  small  scale,  is  seen  in  the  apparatus  of 
a  kitchen  fire-place. 


MECHANICS.  209 

294.  When  one  raises  a  ladder  from  the  ground  by  one  of  the 
ower  rounds,  the  ladder  becomes  a  lever  of  the  third  kind,  the 

power  being  applied  between  the  weight  and  the  prop.  Since 
in  all  the  mechanical  powers,  the  power  and  weight  have  equal 
momenta,  and  since,  in  the  third  kind  of  lever,  the  weight  has 
more  velocity  than  the  power,  the  power  is  as  much  greater  than 
the  weight  as  the  velocity  with  which  it  moves  is  less.  The  dif- 
ficulty experienced  in  raising  a  ladder  from  the  ground  by  taking 
hold  of  the  lowest  round,  or  of  shutting  a  door  by  applying  the 
hand  to  the  side  next  to  the  hinges,  shows  the  mechanical  disad- 
vantage under  which  a  lever  of  this  kind  acts.  Yet  it  is  very 
useful  in  cases  when  it  is  required  to  give  great  velocity  to  the 
body  moved.  Sheep-shears  consist  of  two  levers  of  this  kind 
united.  Here  the  whole  force  required  is  so  small  that  to  save 
it  is  of  no  consequence,  while  so  soft  and  flexible  a  substance  as 
wool,  requires  the  shears  to  be  moved  with  considerable  velocity. 
A  pair  of  tongs  is  composed  in  the  same  manner ;  and  therefore 
it  is  only  a  small  weight  that  we  can  lift  with  them,  especially 
when  the  legs  are  long. 

295.  One  of  the  most  remarkable  applications  of  the  third 
kind  of  lever,  is  in  the  bones  of  animals.     These  are  levers,  the 
joints  are  the  fulcrums,  and  the  muscles  are  the  powers.     The 
muscles  are  endowed  with  a  strong  power  of  contraction,  by 
which  the}r  are  made  to  pull  upon  a  tendon  or  cord,  which  is  in- 
serted in  the  bone  near  the  fulcrum.     Thus,  the  fore-arm  moves 
on  the  joint  near  the  .elbow  as  a  fulcrum,  a  little  below  which  is 
inserted  a  tendon,  connected  with  a  muscle  between  the  elbow 
and  the  shoulder  which  gives  it  motion.     The  arrangement  may 
be  well  represented  by  attaching  a  small  cord  to  one  of  the  legs 
of  a  pair  of  .tongs,  near  the  joint.     It  will  require  a  considerable 
force  to  lift  the  leg  by  pulling  at  the  string,  especially  if  the 
string  be  pulled  in  a  direction  nearly  parallel  with  the  leg,  as  it 
ought  to  be,  since  the  tendon  which  lifts  the  fore-arm,  acts  in 
such  a  direction  with  respect  to  the  arm.     The  muscles  there- 
fore act  in  moving  the  bones  under  a  double  mechanical  disad- 
vantage, their  force  being  applied  both  obliquely  and  very  near 
the  fulcrum.     The  force  which  the  muscles  exert  in  raising  a 
weight  held  in  the  palm  of  the  hand,  is  enormous,  as  will  be 
comprehended  from  the  following  illustration.     Let  AB  (Fig. 

Fig.  143. 


143)  represent  the  fore-arm  moving  on  the  elbow  joint  at  A,  and 
having  the  tendon  inserted  at  C,  which  we  will  suppose  to  be  ten 
times  nearer  to  A  than  B  is  to  A.  Consequently,  a  weight  of  10 

27 


210  NATURAL   PHILOSOPHY. 

pounds  at  B,  would  require  a  force  at  C,  acting  directly  upward, 
of  100  pounds.  But  the  force  of  the  tendon  does  not  act  directly 
upward  in  the  direction  of  CD,  but  very  obliquely,  as  in  the  di- 
rection of  CE,  of  which  the  part  EA  only  can  contribute  to  sup- 
port the  weight.  Suppose  this  part  to  equal  T^th  of  the  whole 
force  CE,  and  it  follows  that  the  muscular  force  exerted  to  raise 
a  weight  of  ten  pounds  in  the  palm  of  the  hand,  would,  were  it 
to  act  without  any  mechanical  disadvantage,  be  sufficient  to  raise 
a  weight  of  1000  pounds.  Yet  Dr.  Young  informs  us,  that  a  few 
years  ago  there  was  a  person  at  Oxford,  who  could  hold  his  arm 
extended  for  half  a  minute,  with  half  a  hundred  weight  hanging 
to  his  little  finger.* 

But  by  giving  to  the  muscle  the  position  it  has,  the  greatest 
possible  compactness  of  structure  is  obtained,  while,  by  making  it 
act  so  near  the  fulcrum,  what  is  lost  in  force,  is  gained  in  veloci- 
ty ;  and  while  the  power  acts  through  a  small  space,  the  hands 
are  moved  quickly  through  a  great  distance.  In  consequence  of 
the  dominion  which  man  can  gain  over  the  stronger  animals, 
and  especially  over  the  great  powers  of  Nature,  he  has  little 
occasion  to  exert  great  strength  with  his  naked  hands :  the  ce- 
lerity of  their  movements  is  to  him  a  far  more  important  en- 
dowment. 

WHEEL   WORK. 

296.  When  a  lever  is  applied  to  raise  a  weight,  or  to  over- 
come a  resistance,  the  space  through  which  it  acts  at  one  time  is 
small,  and  the  work  must  be  accomplished  by  a  succession  of 
short  and  intermitting  efforts.  The  common  lever  is,  therefore, 
used  only  in  cases  where  weights  are  required  to  be  raised 
'through  small  spaces.  When  a  continuous  motion  is  required, 
as  in  raising  ore  from  the  mine,  or  in  weighing  the  anchor  of  a 
vessel,  some  contrivance  must  be  adopted  to  remove  the  inter- 
mitting action  of  the  lever  and  render  it  continual.  The  wheel 
and  axle,  in  its  various  forms,  fully  answers  this  purpose.  It 
may  be  considered  as  a  revolving  lever. 

In  numerous  forms  of  the  wheel  and  axle,  the  weight  is  ap- 
plied by  a  rope  coiled  upon  the  axle  ;  but  the  manner  in  which 
the  power  is  applied  is  very  .various,  and  not  always  by  means  of 
a  rope.  The  circumference  of  a  wheel  sometimes  carries  pro- 
jecting pins  as  in  Fig.  80,  to  which  the  hand  is  applied  to  turn 
the  machine.  An  instance  of  this  occurs  in  the  wheel  used  in 
the  steerage  of  a  vessel. .  In  the  common  windlass  the  power  is 
applied  by  means  of  a  winch,  which  corresponds  to  the  radius 
of  a  wheel.  (See  Fig.  97.)  The  axis  is  sometimes  placed  in  a 
vertical  position,  and  turned  by  levers  moved  horizontally.  The 
capstan  of  a  ship  (Fig.  144)  is  an  example  of  this.  Levers  an- 

*  Young's  Lectures  on  Nat.  Phil.,  I,  129. 


MECHANICS. 


211 


Fig.  144. 


swering  to  the  radii  of  a  wheel,  are  in- 
serted in  holes  mortised  in  the  axis, -and 
turned  by  several  men  working  together. 
In  some  cases,  as  in  the  treadmill,  the 
wheel  is  turned  by  the  weight  of  ani- 
mals walking  on  the  circumference,  with 
a  motion  like  that  of  ascending  a  steep 
hill. 


297.  The  power  of  the  wheel  and  axle  being  expressed  by  the 
number  of  times  the  diameter  of  the  axle  is  contained  in  that  of 
the  wheel,  there  are  obviously  two  ways  by  which  this  power 
may  be  increased;  either  by  increasing  the  diameter  of  the  wheel, 
or  by  diminishing  that  of  the  axle.  In  cases  where  great  power 
is  required,  each  of  these  methods  is  attended  with  practical  in- 
convenience and  difficulty.  If  the  diameter  of  the  v.heel  is  con- 
siderably enlarged,  the  machine  will  become  unwieldy,  and  the 
power  will  work  through  an  unmanageable  space.  If,  on  the 
other  hand,  t'he  power  of  the  machine  is  increased  by  reducing 
the  thickness  of  the  axis,  the  strength  of  the  axis  will  become 
insufficient  for  the  support  of  that  weight,  the  magnitude  of 
which  had  rendered  the  increase  of  the  power  of  the  machine 
necessary.  To  combine  the  requisite  strength  with  moderate 
dimensions  and  great  mechanical  power,  is  therefore  impracti- 
cable, in  the  ordinary  form  of  the  wheel  and  axle.  This  has, 
however,  been  accomplished  by  giving  different  thicknesses  to 
different  parts  of  the  axle,  and  carrying  a  rope,  which  is  coiled 
on  the  thinner  part,  through  a  wheel  attached  to  the  weight, 
and  coiling  it  in  the  opposite  direction  on  the  thicker  part,  as  in 
Fig.  145.  To  investigate  the  proportion  of  the  power  to  the 


Fig.  145. 


Fig.  146. 


weight  in  this  case,  let  Fig.  146  represent  a  section  of  the  ap- 
paratus at  right  angles  to  the  axis.  The  weight  is  equally  sus- 
pended by  the  two  parts  of  the  rope  S  and  S',  and  therefore  each 
part  is  stretched  by  a  force  equal  to  half  the  weight.  The 
momentum  of  the  force  which  stretches  the  rope  S,  is  half  the 
weight  multiplied  by  the  radius  of  the  thinner  part  of  the  axis. 
This  force  being  on  the  same  side  of  the  center  with  the  power, 


212  NATURAL   PHILOSOPHY. 

co-operates  with  it  in  supporting  the  force  which  stretches  S', 
and  which  acts  on  the  other  side  of  the  center.  Now  the  mo- 
menta of  P  and  S  together,  must  be  equal  to  the  momentum  of 
S',  (Art.  21,)  and  therefore  if  P  be  multiplied  by  the  radius  of 
the  wheel,  and  added  to  half  the  weight  multiplied  by  the  radius 
of  the  thinner  part  of  the  axis,  we  shall  obtain  a  sum  equal  to 
half  the  weight,  multiplied  by  the  radius  of  the  thicker  part  of 
the  axis.  Hence  the  power  multiplied  by  the  radius  of  the  wheel, 
is  equal  to  half  the  weight  multiplied  by  the  difference  of  the 
radii  of  the  thicker  and  thinner  parts  of  the  axis.* 

298.  A  wheel  and  axle  constructed  in  this  manner,  is  equiva- 
lent to  an  ordinary  one,  in  which  the  wheel  has  the  same  radius, 
and  whose  axis  has  a  radius  equal  to  half  the  difference  of  the 
radii  of  the  thicker  and  thinner  parts.  f     The  power  of  the  ma- 
chine is  expressed  by  the  ratio  which  the  radius  of  the  wheel 
bears  to  half  the  difference  of  these  radii  ;  and  therefore  this 
power,  when  the  diameter  of  the  wheel  is  given,  does  not,  as  in 
the  ordinary  wheel  and  axle,  depend  on  the  smallness  of  the  axle, 
but  on  the  smallness  of  the  difference  of  the  thinner  and  thicker 
parts  of  it.     The  axle  may,  therefore,  be  constructed  of  such  a 
thickness  as  to  give  it  all  the  requisite  strength,  and  yet  the  dif- 
ference of  the  diameters  of  its  different  parts  may  be  so  small  as 
to  give  it  all  the  requisite  power.J 

We  see  here  strikingly  exemplified  the  principle,  that  the 
weight  sustained  by  a  given  power  may  be  increased  as  its  velo- 
city is  diminished.  By  inspecting  Fig.  146,  it  will  be  seen  that 
the  string  connected  with  the  thinner  part  of  the  axle  unwinds, 
while  that  connected  with  the  thicker  part  winds  up,  by  which 
means  the  ascent  of  the  weight  may  be  rendered  slow  in  any 
degree,  and  a  proportionally  greater  quantity  of  matter  may  be 
added  to  balance  the  constant  momentum  of  the  power. 

299.  It  is  sometimes  desirable  to  make  a  variable  power  pro- 
duce a  constant  force.     This  may  be  done  by  making  its  velocity 
increase  as  its  intensity  diminishes.     We  have  an  example  of  this 
in  the  reciprocal  action  between  the  main-spring  and  fusee  of  a 
watch.     (Fig.  147.)     The  main-spring  is  coiled  up  in  the  box  A, 
and  is  connected  with  the  fusee  B  by  a  chain.     When  the  watch 
is  first  wound  up,  the  spring  acts  with  its  greatest  intensity, 
but  then  as  the  wheel  B  turns,  it  uncoils  with  the  least  velocity  ; 
but  on  account  of  the  varying  diameters  of  the  wheels  of  the 

•          —  —  -  '  - 

*LetW=weight.    P=power.    R=radius  of  the  wheel. 

r=radius  of  the  thicker  part.    r'=radius  of  the  thinner  part. 
Then  PxR+iWXr'=$WXr, 


.. 

t  PxR=iW(r-r')=Wxi(r-r'). 
t  Lardner's  El.  Mech.  p.  181. 


213 


fusee,  the  velocity  is  continually  increased  as  the  intensity  of  the 
spring  is  diminished.  In  a  similar  manner  a  varying  weight  may 
he  moved  by  a  constant  power. 

Communication  of  Motion  by  Wheel-work. 

300.  Motion  may  be  transmitted  by  means  of  wheel-work  in 
several  different  methods,  the  principal  of  which  are,  the  friction 
of  the  circumference  of  one  wheel  upon  that  of  another — the 
friction  of  a  band — and  the  action  of  teeth. 

One  wheel  is  sometimes  made  to  turn  another,  by  the  mere 
friction  of  the  two  circumferences.  If  the  surfaces  of  both  were 
perfectly  smooth,  so  that  all  friction  was  removed,  it  is  obvious 
that  either  would  slide  over  the  surface  of  the  other,  without 
communicating  motion  to  it.  But,  on  the  other  hand,  if  there 
were  any  asperities,  however  small,  upon  their  surfaces,  they 
would  become  mutually  inserted  among  each  other,  and  neither 
the  wheel  nor  axle  could  move  without  causing  the  asperities  on 
its  edge  to  encounter  those  which  project  from  the  surface  of  the 
other ;  and  thus  both  wheel  and  axle  would  move  at  the  same 
time.  Hence  if  the  surfaces  of  the  wheel  and  axle  are  by  any 
means  made  rough,  and  pressed  together  with  sufficient  force, 
the  motion  of  either  will  turn  the  other,  provided  the  load  or  re- 
sistance be  not  greater  than  the  force  necessary  to  break  off  these 
small  projections  which  produce  friction. 

In  some  cases  where  great  power  is  not  required,  motion  is 
communicated  in  this  way  through  a  train  of  wheel- work,  by 
rendering  the  surfaces  of  the  wheel  and  axle  rough,  either  by 
facing  them  with  buff  leather,  or  with  wood  cut  across  the  grain. 
The  communication  of  motion  between  wheels  and  axles  by 
friction  has  the  advantage  of  great  smoothness  and  evenness, 
and  of  proceeding  with  little  noise  ;  but  this  method  can  be  used 
only  in  cases  where  the  resistance  is  not  very  considerable,  and 
therefore  it  is  seldom  adopted  in  works  on  a  large  scale.  Dr. 
Gregory  mentions  an  instance  of  a  saw-mill  at  Southampton, 
where  the  wheels  act  upon  each  other,  by  the  contact  of  the  end 
grain  of  the  wood.  The  machinery  makes  very  little  noise  and 
wears  well,  having  been  used  not  less  than  twenty  years.* 

301.  Wheel  work  is  extensively  moved  by  the  friction  of  a  band. 

*  Gregory's  Mech.  II,  537.— Lardner's  El.  Mech.  183. 


214  NATURAL   PHILOSOPHY. 

When  a  round  cord  is  used,  any  degree  of  friction  may  be  pro- 
duced, by  letting  the  cord  run  in  a  sharp  groove  at  the  edge  of 
the  wheel.  When  a  strap  or  flat  band  is  used,  its  friction  may 
be  increased  by  increasing  its  width.  The  surface  at  the  cir- 
cumference of  a  wheel  which  carries  a  flat  band,  should  not  be 
exactly  cylindrical,  but  a  little  convex,  in  which  case  if  the  band 
inclines  to  slip  off  at  either  side,  it  returns  again  by  the  tighten- 
ing of  its  inner  edge,  as  may  be  seen  in  a  turner's  lathe.  When 
wheels  are  connected  in  the  Fig-  148- 

shortest  manner  by  a  band,  they 
move  in  the  same  direction:  if 

the   band  be  crossed,  they  will  B((  • })  ((       o.       HA 

move  in  opposite  directions.* 
(Fig.  148.)  Wheels  are  some- 
times turned  by  chains  instead 
of  straps  or  bands,  and  are  then 
called  rag  wheels.  The  chains  B 
lay  hold  upon  pins,  or  enter  into 
notches,  in  the  circumference  of 
the  wheels,  so  as  to  cause  them 
to  turn  simultaneously.  They  are  used  where  it  is  necessary 
that  the  velocities  should  be  uniform,  and  where  great  resistance 
is  to  be  overcome,  as  in  locomotive  steam  engines,  chain  water 
wheels,  &c.f 

302.  But  the  most  common  mode  of  moving  wheel-work,  is 
by  means  of  teeth  cut  in  the  circumference  of  the  wheels.  The 
wheels  of  necessity  turn  in  opposite  directions.  The  connexion 
of  one  toothed  wheel  with  another  is  called  gearing.  In  the 
formation  of  teeth,  very  minute  attention  must  be  given  to  their 
figure,  in  order  that  motion  may  be  communicated  from  one 
wheel  to  another,  without  rubbing  or  jarring.  If  the  teeth  are 
ill  matched,  as  in  Fig.  149,  when  the  tooth  A  comes  in  contact 

Fig.  149.  Fig.  150. 


with  B,  it  acts  obliquely  upon  it,  and  as  it  moves,  the  corner  of 


*  Bigelow's  Technology,  p.  229. 


t  Ib.  p.  230. 


MECHANICS.  215 

B  slides  upon  the  plane  surface  of  A  in  such  a  manner  as  to  pro- 
duce much  friction,  and  to  grind  away  the  side  of  A,  and  the  end 
of  B.  As  they  approach  the  position  CD,  they  sustain  a  jolt  the 
moment  their  surfaces  come  into  full  contact ;  and  after  passing 
the  position  CD,  the  same  scraping  and  grinding  effect  is  pro- 
duced in  the  opposite  direction,  until  by  the  revolution  of  the 
wheels  the  teeth  become  disengaged.  To  avoid  these  evils,  the 
surfaces  of  the  teeth  are  frequently  curved  so  as  to  roll  on 
each  other  with  as  little  friction,  and  with  as  uniform  force  and 
velocity  as  possible.  (Fig.  150.)  Much  pains  and  skill  have 
been  bestowed  on  this  subject  by  mathematicians,  with  the  view 
of  ascertaining  the  kinds  of  curves  which  fulfil  these  purposes 
best.* 

Regulation  of  Velocity  by  Wheel-work. 

303.  Wheel-work  serves  the  purpose,  not  only  of  forming  a 
convenient  communication  of  motion  between  the  power  and 
the  weight,  but  also  of  regulating  its  velocity.  Thus,  when  the 
connection  is  formed  by  means  of  a  band,  as  in  Fig.  148,  the 
velocity  of  the  wheel  B,  that  carries  the  weight  or  sustains  the 
pressure,  may  be  altered  at  pleasure,  by  altering  the  ratio 
between  the  diameters  of  the  two  wheels.  If  the  diameters  are 
equal,  the  wheels  will  revolve  with  equal  velocity ;  if  A  re- 
mains the  same,  while  the  diameter  of  B  is  diminished,  the 
velocity  of  B  will  be  increased  in  the  same  ratio  ;  or  if  B  remains 
the  same,  while  the  diameter  of  A  is  changed,  the  velocity  of  B 
will  be  changed  in  the  same  manner,  f  We  see  familiar  ex- 
amples of  the  application  of  this  principle  in  the  common  spin- 
ning wheel,  and  the  turner's  lathe.  In  the  spinning  wheel  a 
band  passes  round  a  large  wheel  and  a  small  one  called  a  spool, 
ha\4ng  the  spindle  for  its  axis  ;  and  in  consequence  of  the  great 
disparity*  in  the  size  of  the  wheels,  a  great  velocity  is  given  to 
the  spindle  by  a  comparatively  slow  revolution  of  the  wheel. 
In  a  turner's  apparatus,  machinery  for  spinning  cotton,  and  the 
like,  a  large  hollow  cylinder  or  drum,  is  fixed  horizontally,  which 
is  kept  revolving  by  the  moving  power,  and  from  which,  motion 
is  conveyed  by  bands  to  lathes,  spindles,  &c.,  to  which  any 
required  velocity  is  given,  by  altering  the  diameter  of  the  small 
wheel  that  is  connected  with  them  and  turns  them.  Sometimes 
a  change  of  velocity  is  effected  by  making  the  drum  of  a  conical 
shape,  and  then  the  velocity  imparted  to  the  lathe  or  the  spindle, 
will  be  greater  or  less,  according  as  the  band  proceeds  from  the 
larger  or  smaller  part  of  the  drum. 

*  See  Blake  on  the  form  of  the  teeth  of  cog  wheels,  Am.  Journ.  Sci.,  VIII,  86. 

t  This  would  be  accurately  true,  in  case  the  band  did  not  slip  or  slide ;  but  since 
it  usually  does  slide  more  or  less,  the  velocity  of  the  driven  wheel  is  commonly  a 
little  less  in  proportion,  than  that  of  the  wheel  which  drives  it. — Bigelow,  El.  Tech 
p.  229. 


216  NATURAL   PHILOSOPHY. 

304.  A  more  exact  method  of  regulating  the  velocity  of  mo- 
tion, is  by  means  of  wheels  and  pinions*     An  example  of  this 
kind  is  seen  in  Fig.   151,  where  A,  B,  C,  are  three  wheels,  and 
a,  b,  c,  are  the  corresponding  pinions.     As  the  leaves  of  the  pin- 
ions successively  pass  between  Fig.  151. 

the  teeth  of  the  wheel,  the  di- 
visions of  the  two  circumferen- 
ces must  correspond  to  each 
other,  and  the  number  of  teeth 
in  the  wheel  will  be  as  much 
greater  than  in  the  pinion,  as  the 
circumference  of  the  wheel  is 
greater  than  that  of  the  pinion. 
Therefore  it  follows,  that  the 
number  of  teeth  in  a  wheel,  and 
of  leaves  in  the  pinion  that  acts 
upon  it,  expresses  the  ratio  of  the 
circumference  or  radius  of  the  wheel  to  that  of  the  pinion. 
Hence,  in  an  equilibrium,  the  power  multiplied  by  the  product  of 
the  numbers  expressing  the  amount  of  teeth  in  all  the  wheels 
respectively,  is  equal  to  the  weight  multiplied  by  the  product  of 
the  several  numbers  denoting  the  leaves  in  each  of  the  pinions. 
(Art.  117.) 

It  is  further  evident  that  the  velocity  of  the  wheel  and  that  of 
the  pinion  connected  with  its  circumference,  will  be  inversely 
as  the  number  of  teeth  in  each.  Thus  in  Fig.  151,  if  the  pinion 
a  has  10  teeth,  and  the  wheel  B  has  100,  a  will  move  ten  times 
as  fast  as  B.  For  the  same  reason  b  will  move  ten  times  as  fast 
as  C  ;  so  that,  in  this  arrangement,  the  power  moves  with  100 
times  the  velocity  of  the  weight.  By  varying  the  ratio  between 
the  number  of  leaves  in  the  pinion,  and  the  number  of  teeth  in 
the  wheel  with  which  it  is  connected,  we  may  vary  the  velocity 
of  any  wheel  at  pleasure. 

305.  A  familiar  instance  of  this  is  afforded  in  the  mechanism 
of  a  common  clock.     A  pendulum  by  falling  gains  a  quantity  of 
motion  sufficient  to  carry  it,  on  the  other  side,  to  the  same  height 
as  that  from  which  it  fell ;  and  were  it  not  for  the  resistance  of 
the  air  and  the  impediments,  a  pendulum  when  once  set  in  mo- 
tion would  continue  to  vibrate  by  its  own  inertia,  (Art.  19,)  and 
would  thus  afford,  without  the  aid  of  any  machinery,  an  exact 
measure  of  time.     But,  in  order  to  continue  its  vibrations,  some 
small  force  must  be  applied  to  it  to  compensate  for  the  loss  of 
motion  from  friction  and  resistance.     This  force  is  applied  to  the 

*  Pinions  are  smaller  wheels  acting  on  the  circumferences  of  larger.  The  teeth  of 
a  pinion  are  called  leaves.  They  are  most  commonly  raised  on  the  axis  of  one  wheel, 
and  form  the  communi  sation  between  that  wheel  and  the  next  in  the  series. 


MECHANICS. 


217 


K 


pendulums  of  clocks  by  the  weight,  and  an  analogous  force  is 
supplied  to  the  balance  wheel  of  watches  and  chronometers  by 
springs.  In  Fig.  152,  let  AB  be  a  wheel  Fig.  152. 
having  30  teeth,  and  let  N,  M,  be  a  pendu- 
lum, connected  with  the  wheel  by  the  pallets 
I,  K  ;  and  to  the  axis  a,  let  a  weight  be  hung. 
It  is  evident  that  this  weight,  were  there 
nothing  to  arrest  it,  would  descend  by  the 
force  of  gravity  with  accelerated  velocity.  It 
endeavors  thus  to  descend,  and  hence  exerts 
the  required  force  on  the  pallets  of  the  pendu- 
lum. For,  every  time  the  pendulum  performs 
a  double  vibration,  (returning  to  the  same 
point  from  which  it  set  out,)  a  tooth  of  the 
wheel  escapes,*  and  the  wheel  runs  down 
until  the  next  tooth  strikes  upon  the  pallet, 
and  thus  gives  it  the  impulse  which  is  necessary 
to  keep  up  the  vibrations. 

It  would  seem  therefore  that,  for  beating 
seconds,  only  a  single  wheel  is  necessary ;  nor  would  any  more 
be  absolutely  indispensable ;  but  in  this  case  the  weight  would 
descend  so  fast,  as  soon  to  reach  the  floor,  and  the  clock  would 
require  to  be  wound  up  again  every  few  minutes.  Hence  a 
series  of  wheels  are  interposed  between  the  pendulum  and  the 
weight,  by  which  the  descent  of  the  latter  is  retarded  upon  the 
principle  explained  in  Art.  304,  and  the  descent  of  the  weight  is 
slower  in  proportion  as  the  series  is  more  extensive.  In  cheap 
clocks,  as  some  of  those  made  with  wooden  wheels,  the  series  is 
short,  or  the  number  of  wheels  employed  for  retarding  the  descent 
of  the  weight  is  small,  and  such  clocks  require  frequent  winding 
up  ;  but  in  clocks  of  finer  workmanship,  a  greater  number  of 
wheels  is  interposed,  and  such  clocks  require  to  be  wound  up 
less  frequently.  Many  go  eight  days,  and  some  are  made  to  go 
a  whole  year  without  winding. 


Wheel  Carriages. 

306.  When  a  loaded  carriage  is  moving  on  a  horizontal  plane, 
free  of  obstacles,  the  resistance  to  be  overcome  does  not  consist 
of  the  weight  of  the  load,  directly,  but  of  the  friction  occasioned 
by  the  weight.  For,  since  the  weight  acts  in  a  direction  perpen- 
dicular to  the  plane,  it  cannot  oppose  the  motion  of  the  carriage 
in  a  direction  parallel  to  the  plane.  Nor  would  increasing  the 
weight  to  any  extent,  make  any  difference,  were  it  not  that  we 
should  thus  increase  the  friction,  which  (as  will  be  explained 
more  fully  hereafter)  is  proportioned  to  the  weight. 


*  Hence  this  wheel  is  called  the  scapement. 
28 


218  NATURAL   PHILOSOPHY. 

When  a  carriage  wheel  is  made  to  slide  on  the  ground,  (as 
when  a  wheel  is  locked,)  the  whole  amount  of  the  friction  is  en- 
countered without  bringing  in  to  our  aid  any  mechanical  advan- 
tage ;  but  when  a  wheel  turns  on  its  axle,  the  friction  is  trans- 
ferred from  the  ground  to  the  axle,  and  each  spoke  of  the  wheel 
successively  becomes  a  lever,  turning  on  the  ground  as  a  fulcrum, 
while  the  power  or  force  of  the  team  is  exerted  on  the  end  next 
to  the  axle.  By  thus  transferring  th~  friction  from  the  ground  to 
the  axle,  each  spoke,  in  its  turn,  is  F-  153 

made  to  aid  in  overcoming  that  fric- 
tion.    Thus,  in  Fig.  153,  let  C  be 

the  axle,  CP  the  line  of  draught,  //^^  |,|  //^\  ...---  S 
and  R  the  point  where  the  wheel 
touches  the  plane.  The  force  ap- 
plied in  the  direction  CP,  acts  on 
CR  at  C,  and  turns  it  on  its  fulcrum 
at  R.  This  is  the  force  by  which 
the  wheel  is  made  to  advance.  But 
the  friction  on  the  axle  atC,  reacts  in 
the  opposite  direction,  having  a  leverage  equal  only  to*4he  ra- 
dius of  the  axle,  while  the  power  which  overcomes  this,  has  a 
leverage  equal  to  the  radius  of  the  wheel.  Hence,  in  the  wheel, 
there  is  a  mechanical  advantage  gained  in  overcoming  the  fric- 
tion, in  the  ratio  of  the  radius  of  the  wheel  to  the  radius  of  the 
axle.  Moreover,  the  axle  may  be  made  of  such  materials,  and 
lubricated  with  such  substances,  as  to  render  the  actual  amount 
of  friction  much  less  than  it  would  be  were  the  wheel  made  to 
slide  on  the  ground. 

307.  But  wheels  have  another  important  advantage,  namely,  in, 
overcoming  obstacles;  in  which  case  they  act  on  the  principle  of 
the  bent  lever. 

Thus  let  A  be  an  obstacle,  as  a  stone  for  example.  From  A  let 
fall  the  perpendiculars  AN,  AM,  upon  GR,  CP,  and  conceive  MAN 
to  be  a  bent  lever,  turning  on  A  as  a  fulcrum,  the  power  being 
applied  at  M  in  the  direction  CP,  and  the  weight  resting  on  N, 
which  supports  the  center  of  gravity.  Now,  the  mechanical 
advantage  gained,  will  be  in  the  ratio  of  MA  to  NA.  It  will 
therefore  be  increased  (and  of  courso  the  force  necessary  to  over- 
come the  obstacle  be  diminished)  as  the  point  A  is  nearer  to  R ; 
and  the  mechanical  advantage  will  be  lessened  as  the  point  A  re- 
cedes from  R.  Whqp  the  obstacle  is  so  large  as  to  make  AM  only 
equal  AN,  then  no  mechanical  advantage  is  gained,  but  the  whole 
weight  of  the  load  must  be  lifted  by  the  former  ;  and  when  AM 
becomes  less  than  AN,  the  wheel  involves  a  mechanical  disadvan- 
tage, and  the  difficulty  of  carrying  the  wheel  over  the  obstacle 
becomes  very  great.  It  is  further  obvious  that  large  wheels  have 
the  mechanical  advantage,  both  as  regards  overcoming  the  fric- 


MECHANICS.  219 

tion,  and  overcoming  obstacles,  in  a  higher  degree  than  small 
wheels,  since  these  afford  a  greater  leverage  than  the  others  on 
account  of  the  increased  length  of  the  spokes.  But  in  practice 
very  large  wheels  cannot  be  employed,  since  they  would  be  either 
weak  or  too  heavy,  and  the  increased  height  of  the  axle  would 
carry  the  center  of  gravity  too  high,  and  enhance  the  danger  of 
upsetting.  The  difficulty  of  turning  might  also  render  unusually 
large  wheels  ineligible  ;  and  the  axle  might  be  raised  so  high,  as 
to  make  the  horse  draw  obliquely  downward  and  increase  the 
pressure  on  the  ground,  whereas  the  line  of  draught  ought  to  be 
so  adjusted  as  to  lighten  that  pressure,  especially  where  the  road 
is  soft  and  yielding. 

When  a  wheel  sinks  below  the  surface,  the  force  is  rendered 
strikingly  inefficacious  from  several  causes.  The  fulcrum  on 
which  each  spoke  successively  turns  gives  way,  and  diminishes 
greatly  the  mechanical  advantage  otherwise  gained  by  transfer- 
ring the  friction  from  the  ground  to  the  axle,  as  before  explained. 
Likewise,  the  mud  or  sand  into  which  the  wheel  has  sunk,  op- 
poses in  front  of  the  wheel  an  obstacle  like  that  represented  at 
A  in  Fig.  153,  while  the  fulcrum  on  which  the  bent  lever  turns 
in  the  effort  to  lift  the  wheel  over  the  obstacle  gives  way  as  in 
the  other  case,  and  a  great  part  of  the  mechanical  advantage  is 
lost.  From  these  considerations,  it  is  easy  to  understand  the 
reason  of  the  superior  advantages  of  hard  and  smooth  roads. 

308.  The  line  of  draught  should  not  be  horizontal,  but  inclined 
upward  toward  the  breast  of  the  horse,  in  an  angle  not  less  than 
15  degrees  with  the  horizon.     This  brings  the  strain  nearly  at 
right  angles  with  the  collar,  whereas  a  horizontal  draught  lifts  the 
collar  upward,  by  which  the  force  is  wasted  and  the  animal  is 
choked.*     The  angle  of  draught,  however,  should  be  less  than 
the  above  when  the  road  is  very  smooth.     The  general  rule  is, 
that  the  angle  should  be  the  same  as  the  inclination  of  a  hill, 
down  which  the  carriage  would  roll  spontaneously.    Consequent- 
ly, in  smooth  Macadamized  roads,  the  line  of  draught  should  be 
at  a  small  angle,  and  on  railways  nearly  horizontal.! 

309.  The  effect  of  suspending  a  carriage  on  springs,  is  to 
equalize  the  motion  by  causing  every  change  to  be  more  gradu- 
ally communicated  to   it,  and  to  obviate  shocks.     Springs  are 
not  only  useful  for  the  convenience  of  passengers,  but  they  also 
diminish  the  labor  of  draught ;  for  whenever  a  wheel  strikes  a 
stone,  it  rises  against  the  pressure  of  the  spring,  in  many  cases 
without  materially  disturbing  the   load,  whereas  without  the 
spring,  the  load,  or  a  part  of  it,  must  rise  with  every  jolt  of  the 
wheel,  and  will  resist  the  change  of  place  with  a  degree  of  inertia 

*  Fuller  on  Wheel  Carriages. 

•f  Moseley's  Mechanics  applied  to  the  Arts. 


220  NATURAL   PHILOSOPHY. 

proportionate  to  the  weight,  and  the  suddenness  of  the  percus- 
sion. Hence  springs  are  highly  useful  in  baggage  wagons,  and 
other  vehicles  used  for  heavy  transportation.* 

A  pair  of  horses  draw  more  advantageously  abreast  than  when 
one  is  harnessed  before  the  other.  In  the  latter  case,  the  forward 
horse,  being  attached  to  the  ends  of  the  shafts,  draws  in  a  line 
nearly  horizontal ;  consequently  he  does  not  act  with  his  whole 
force  upon  the  load,  and  moreover  expends  a  part  of  his  force  in 
a  vertical  pressure  on  the  back  of  the  other  horse. 


THE   PULLEY. 


310.  Pulleys  are  divided  into  FIXED  and  MOVABLE.  In  the 
fixed  pulley,  as  has  been  demonstrated  in  Art.  120,  no  mechani- 
cal advantage  is  gained,  but  its  use  consists  in  furnishing  a  con- 
venient mode  of  changing  the  direction  of  the  power.  Thus,  it 
is  far  more  convenient  to  raise  a  bucket  from  a  well  by  drawing 
downward,  as  is  the  case  when  the  rope  passes  over  a  fixed 
pulley  above  the  head,  than  by  drawing  upward  leaning  over 
the  well.  By  means  of  the  pulley,  great  facilities  are  afforded 
for  managing  the  rigging  of  a  ship.  The  sails  at  mast  head 
can  be  easily  raised  while  the  hands  stand  upon  the  deck, 
whereas,  without  the  aid  of  ropes  and  pulleys,  the  same  force 
removed  to  mast  head  would  operate  under  very  great  disad- 
vantages. Similar  facilities  are  afforded  by  this  kind  of  appa- 
ratus for  raising  heavy  weights,  as  boxes  of  merchandise,  or 
Fig.  154. 


Bigelow,  El.  Tech.,  p.  200. 


MECHANICS.  221 

heavy  blocks  of  stone  in  building.  Fig.  154,  represents  a  con- 
venient method  of  elevating  large  masses  of  stone  in  building  a 
high  tower,  as  a  lighthouse  or  a  monument.  The  crane  at  B  en- 
ables the  workmen,  when  the  weight  is  raised,  to  swing  it  round 
to  the  point  where  it  is  to  lie,  or  to  a  platform  near  to  it.  The 
lower  end  of  the  rope  CD,  is  connected  with  a  wheel  and  axle 
in  the  figure,  but  it  is  obvious  that  different  methods  of  applying 
the  power  might  be  adopted,  to  suit  the  convenience  of  the  work- 
men. For  example,  instead  of  the  wheel  and  axle  we  might 
attach  a  horse  or  a  yoke  of  oxen  to  the  rope  CD  ;  or  we  might 
attach  a  long  sweep  to  the  top  of  the  axis,  and  join  a  team  of 
horses  or  cattle  to  the  end  of  it,  and  raise  the  weight  by  driving 
them  round,  as  in  a  common  cider-mill. 

Fire  escapes  sometimes  consist  merely  of  a  pulley  fixed  near 
the  window  of  the  apartment,  around  which  a  rope  may  be 
easily  placed,  having  a  basket  attached  to  the  end.  The  man 
seats  himself  in  the  basket,  grasping,  at  the  same  moment,  the 
rope  on  the  other  side  of  the  pulley,  and  thus  he  lets  himself 
gradually  down. 

311.  The  movable  pulley,  by  distributing  the  weight  into 
separate  parts,  so  that  it  is  supported  at  several  different  points 
at  once,  is  attended  by  a  mechanical  advantage,  proportioned  to 
the  number  of  such  points  of  support.  Movable  pulleys  may 
be  arranged  according  to  several  different  systems,  which  in- 
crease the  efficacy  of  a  given  power  in  different  ratios.*  It  will 
be  observed,  however,  that  the  ascent  of  the  weight  is  in  all 
cases  retarded  in  proportion  as  the  efficacy  of  a  given  power  is 
increased.  It  must  be  further  observed  that  in  using  any  system 
of  movable  pulleys,  the  whole  weight  of  the  pulleys  themselves, 
together  with  the  resistance  occasioned  by  the  rigidity  and  fric- 
tion of  the  rope,  all  act  against  the  power,  and  so  far  lessen  the 
weight  which  it  is  capable  of  raising.  In  the  more  complex  sys- 
tems of  pulleys,  it  is  estimated  that  at  least  two  thirds  of  the 
power  is  expended  on  the  machinery  itself.  On  account  there- 
fore of  the  slowness  of  the  motion  which  the  weight  receives, 
and  the  loss  of  power  from,  the  resistance  of  the  ropes  and  blocks, 
such  systems  of  pulleys  are  seldom  employed.  It  is  only  in  rais- 
ing vast  weights,  such  as  large  ships,  or  great  masses  of  stone 
from  a  quarry,  that  they  are  ever  used.  For  managing  the  rigging 
of  a  ship,  the  combination  usually  employed  consists  of  not  more 
than  two  or  three  movable  pulleys.  From  its  portable  form, 
however,  its  cheapness,  and  the  facility  with  which  it  can  be  ap- 
plied, especially  in  changing  or  modifying  the  direction  of  motion, 
the  pulley  is  one  of  the  most  convenient  and  useful  of  the  me- 
chanical powers. 

*  See  Part  I,  Arts.  122—125. 


222  NATURAL   PHILOSOPHY. 


THE   INCLINED   PLANE. 

312.  The  inclined  plane  becomes  a  mechanical  power  in  con- 
sequence of  its  supporting  a  part  of  the  weight,  and  of  course 
leaving  only  a  part  to  be  supported  by  the  power.     Thus  the 
power  has  to  encounter  only  a  portion  of  the  force  of  gravity  at  a 
time, — a  portion  which  is  greater  or  less,  according  as  the  plane 
is  more  or  less  elevated.     When  a  plane  is  perfectly  horizontal, 
it  sustains  the  entire  pressure  of  a  body  that  rests  on  it ;  that  is, 
the  pressure  on  the  plane  is  equal  to  the  whole  force  of  gravity 
acting  on  the  body.     As  one  end  of  the  plane  is  elevated,  this 
force  is  resolved  into  two,  one  of  which  is  parallel  and  the  other 
perpendicular  to  the  plane.     In  proportion  as  the  plane  is  more 
elevated,  the  part  of  the  force  which  acts  parallel  with  the  plane 
is  increased,  until,  when  the  plane  becomes  perpendicular  to  the 
horizon,  it  no  longer  sustains  any  portion  of  the  weight,  and  the 
latter  descends  with  the  whole  force  of  gravity. 

313.  The  simplest  example  we  have  of  the  application  of  the 
inclined  plane,  is  that  of  a  plank  raised  at  the  hinder  end  of  a 
cart  for  the  purpose  of  rolling  in  heavy  articles,  as  barrels  or 
hogsheads.     The  force  required  to  roll  the  body  on  the  plank, 
setting  aside  friction,  is  as  much  less  than  that  required  to  lift  it 
perpendicularly,  as  the  height  of  the  plane  above  the  ground  is 
less  than  its  length.     Every  one  knows  how  much  the  facility  of 
moving  heavy  loads  is  increased  by  such  means,  and  how  the 
force  required  to  move  them  is  diminished,  by  increasing  the 
length  of  the  plane  while  the  height  remains  the  same.     Long 
inclined  planes,  constructed  of  plank,  are  frequently  employed  in 
building,  especially  where  high  walls  are  built  of  large  masses 
of  stone,  the  materials  being  trundled  upon  the  plane  on  wheel- 
barrows, or  transported  on  heavy  rollers.     It  is  even  supposed, 
that  in  building  the  Pyramids  of  Egypt,  the  huge  masses  of 
stones  were  elevated  on  inclined  planes.     Roads  also,  except 
when  they  are  perfectly  level,  afford  examples  of  this  mechanical 
power.     When  a  horse  is  drawing  a  heavy  load  on  a  perfectly 
horizontal  plane,  what  is  it  that  occasions  such  an  expenditure 
of  force  ?     It  is  not  the  weight  of  the  load,  except  so  far  as  that 
increases  the  friction  ;  for  gravity,  acting  in  a  direction  perpen- 
dicular to  the  horizon,  can  oppose  no  resistance  in  the  direction 
in  which  the  load  is  moving.     The  answer  is,  that  the  force  of 
the  horse  is  expended  chiefly  in  overcoming  friction,  and  the  re- 
sistance of  the  air.     But  when  a  horse  is  drawing  a  load  up  a 
hill,  he  has  not  only  these  impediments  to  encounter,  but  has 
also  to  overcome  more  or  less  of  the  force  of  gravity ;  that  is, 
he  lifts  such  a  part  of  the  load  as  bears  to  the  whole  load  the 
same  ratio,  that  the  perpendicular  height  of  the  hill  bears  to  its 


MECHANICS.  223 

length.  If  the  rise  is  one  foot  in  twenty,  he  lifts  one  twentieth 
of  the  load,  and  therefore  encounters  so  much  resistance  in  ad- 
dition to  those  which  he  had  to  overcome  on  the  horizontal  plane. 
If  the  ascent  were  one  foot  in  four,  and  the  load  were  a  ton, 
the  additional  force  required  above  what  would  be  necessary  on 
level  ground,  would  be  560  pounds. 

314.  Railways  afford  another  striking  exemplification  of  the 
principles  of  the  inclined  plane.     By  means  of  them  the  irregu- 
lar surface  of  a  country,  however  hilly  and  uneven,  is  reduced 
to  horizontal  levels  and  inclined  planes.     These  are  sometimes 
inclined  at  so  slight  an  angle,  that  the  tendency  of  the  cars  down 
the  plane,  is  only  just  sufficient  to  balance  their  friction,  and 
they  would  remain  at  rest  of  themselves  in  any  part  of  the  plane, 
while  a  small  force  would  move  them  either  way.     In  other 
places  the  inclined  planes  are  very  steep  for  a  short  distance  ; 
and  the  cars  ascending  them  are  sometimes  drawn  up  by  means 
of    a  power  (a  steam  engine,  for  example)   stationed  on  the 
summit,  and  sometimes  cars  descending  on  one  side  are  made  to 
draw  up  others  on  the  other  side,  the  two  being  connected  by  a 
chain  or  rope  which  passes  round  a  pulley  on  the  summit.* 

315.  When  railways  first  came  into  use,  the  power  of  one 
horse  was  considered  only  equal  to  a  load  of  10  tons  ;  but  it  is 
now  estimated  that,  upon  a  level  railroad  of  the  best  construc- 
tion with  carriages  of  the  most  perfect  finish,  a  horse-power  is 
equivalent  to  a  load  of  222  tons,  although  an  average  load  is 
considered  to  be  about  16  tons.     The  resistances  to  be  overcome 
have  thus  been  reduced  to  only  ^}F  of  the  weight,  and  may  be- 
safely  taken  at  ¥^ ,  while  upon  the  best  common  roads  it  is 
never  less  than  -^\,  and  is,  in  most  cases,  as  great  as  ^.     The 
advantage  of  a  good  railroad  over  a  common  turnpike,  when 
horses  are  employed,  is  therefore  in  a  tenfold  proportion.     But 
railroads  derive  their  greatest  value  from  the  employment  of 
steam   as  a  moving  power,  as  exhibited  in  the  locomotive.f 
Horses  tire  at  a  moderate  speed,  but  steam  never  tires,  and  is 
therefore  peculiarly  adapted  to  the  transportation  of  passengers, 
where  great  expedition  is  required.     By  means  of  this  force,  a 
speed  of  20  miles  an  hour  is  easily  gained,  and  in  some  extreme 
cases,  it  has  been  pushed  as  high  as  60  miles  an  hour.     But 
such  immense  loads  moving  with  such  great  velocities,  acquire 
a  momentum  that  is  truly  formidable,  and  involves  inevitable 
danger. 

316.  In  slow  motions,  canals  have  some  advantages  over  rail- 
ways.    A  horse  easily  draws  on  a  canal  a  load  of  30  tons ;  and 

*  Strickland's  Reports.  t  Renwick's  Practical  Mechanics. 


224  NATURAL   PHILOSOPHY. 

it  is  said  that  the  employment  of  several  horses  to  a  boat  is  ad- 
vantageous, since  the  weight  drawn  increases  in  a  higher  ratio 
than  the  number  of  horses.  When,  therefore,  heavy  loads,  with 
slow  motions,  are  to  be  transported,  canals  have  an  advantage 
over  railroads  of  two  to  one.*  But  canal  boats  cannot  be  made 
to  move  with  much  speed,  without  injury  to  the  canal  and  great 
loss  of  force  by  the  increased  resistance  of  the  water,  this  being 
augmented  in  proportion  to  the  square  of  the  velocity.  In  rail- 
ways, on  the  other  hand,  the  resistance  arising  from  the  friction 
of  the  wheels  on  the  road,  is  diminished  as  the  velocity  is  in- 
creased, while  the  resistance  of  the  air,  although  increased  by 
an  increase  of  velocity  like  any  other  fluid,  is  so  small  as  to  oc- 
casion no  serious  impediment,  being  at  a  velocity  of  14  miles  per 
hour,  only  about  1  Ib.  on  every  square  foot  of  the  front  of  the 
leading  carriage. 

THE  SCREW. 

317.  When  a  road,  instead  of  ascending  a  hill  directly,  winds 
round  it  to  the  summit,  so  as  to  lengthen  the  inclined  plane,  and 
thus  aid  the  moving  force,  the  inclined  plane  becomes  a  screw. 
In  the  same  manner  a  pair  of  stairs,  winding  around  the  sides 
of  a  cylindrical  tower,  either  within  or  without,  affords  an  in- 
stance of  an  inclined  plane  so  modified  as  to  become  a  screw. 
These  examples  show  the  strong  analogy  which  subsists  between 
these  two  mechanical  powers ;  or  rather,  they  show  that  the 
screw  is  a  mere  modification  of  the  inclined  plane. 

318.  The  screw  is  generally  employed  when  severe  pressure 
is  to  be  exerted  through  small  spaces,  and  is  therefore  the  agent 
in  most  presses.     Being  subject  to  great  loss  from  friction,  (upon 
which  however  its  chief  utility  depends,  as  will  be  shown  here- 
after,) it  usually  exerts  but  a  small  power  of  itself,  but  derives 
its  principal  efficacy  from  the  lever,  or  from  wheel- work,  with 
which  it  is  very  easily  combined.     Thus,  in  Fig.  155,  were  the 
power  applied  directly  to  the  screw,  the  mechanical  advantage 
gained  would  hardly  more  than  compensate  for  the  loss  by  fric- 
tion ;  but  by  means  of  the  lever,  (which  may  be  lengthened  or 
shortened  at  pleasure,)  the  power  is  greatly  increased.     Also,  by 
means  of  the  endless  screw,  Fig.  98,  combined  with  the  wheel 
and  axle,  a  very  powerful  force  may  be  exerted ;  and  as  the 
mechanical  power  of  the  screw  depends  upon  the  relative  mag- 
nitude of  the  circumference  through  which  the  power  revolves, 
and  the  distance  between  the  threads,  (Art.  139,)  it  is  evident 
that,  to  increase  the  efficacy  of  the  machine,  we  must  either  in- 
crease the  length  of  the  lever  by  which  the  power  acts,  or  di- 

*  Renwick's  Practical  Mechanics. 


x.; 

MECHANICS.  225 

minish  the  distance  between  the  threads.  Although,  in  theory, 
there  is  no  limit  to  the  increase  of  the  mechanical  efficacy  by 
these  means,  yet  practical  inconvenience  arises  from  the  great 
space  over  which  a  very  long  lever  traverses.  If,  on  the  other 
hand,  the  power  of  the  machine  is  increased  by  diminishing  the 
distance  between  the  threads,  and  of  course  their  size,  the  thread 
will  become  too  slender  to  bear  a  great  resistance.  The  cases 
in  which  it  is  necessary  to  increase  the  power  of  the  machine, 
being  those  in  which  the  greatest  resistances  are  to  be  over- 
come, the  object  will  evidently  be  defeated,  if  the  means  chosen 
to  increase  that  power,  deprives  the  machine  of  the  strength 
which  is  necessary  to  sustain  the  force  to  which  it  is  to  be  sub- 
mitted.* 

319.  These  inconveniences  are  remedied  by  Hunter's  Screw, 
which,  while  it  gives  to  the  machine  all  the  requisite  strength 
and  compactness,  allows  it  to  have  an  almost  unlimited  degree 
of  mechanical  efficacy.  This  contrivance,  which  is  represented 
in  Fig.  155,  acts  upon  a  principle  similar  Fig.  155. 

to  that  of  the  wheel  and  axle,  as  repre- 
sented in  Fig.  145,  where  the  efficacy  of 
the  power  is  increased  by  diminishing  the 
velocity  of  the  weight,  which  is  accom- 
plished by  making  the  rope  unwind  on  one 
side  while  it  winds  up,  with  somewhat 
greater  speed,  on  the  other  side.  In  the 
case  before  us,  the  screw  is  likewise  com- 
posed of  a  smaller  and  a  larger  thread,  the 
former  turning  upward  while  the  latter 
turns  downward  with  a  little  greater  ve- 
locity, and  consequently  the  screw,  on  the 
whole,  advances  with  the  difference  between  the  larger  and  the 
smaller  threads  ;  and  since  this  difference  may  be  small  to  any 
extent,  so  the  efficacy  of  the  power  may  be  increased  indefinite- 
ly. It  will  be  seen,  however,  that  the  motion  of  such  a  screw  is 
exceedingly  slow.  Thus  in  Fig.  155,  A  descends,  while  B,  play- 
ing in  a  concave  screw  in  A,  ascends  ;  but  the  distance  between 
the  threads  of  A  being  greater  than  the  distance  between  those 
of  B,  the  screw,  on  the  whole,  advances  with  the  difference. 
Suppose  that  A  has  20  threads  in  an  inch,  and  B  21  ;  then  du- 
ring one  revolution,  A  will  descend  through  the  20th,  while  B 
ascends  through  the  21st  part  of  an  inch.  The  compound  screw, 
therefore,  will  advance  through  a  space  equal  to  the  difference : 
that  is,  through  a  space  equal  to  ^-^=^$1  of  an  inch.  This 
small  space  is  therefore,  in  effect,  the  distance  between  two  con- 
tiguous threads  ;  and  the  power  of  the  machine  is,  as  usual,  ex- 

»  Lardner's  El.  Mech.,  p.  220. 
29 


226  NATURAL   PHILOSOPHY. 

pressed  by  the  number  of  times  their  distance  is  contained  in 
the  circumference  described  in  one  revolution  of  the  power. 
For  example,  let  the  circumference  of  the  circle  be  one  foot ; 
then  12H-Tjo=5040=the  weight  or  resistance,  the  power  being 
1  ;  or,  in  other,  words,  the  efficacy  of  the  power  is  increased  five 
thousand  and  forty  times. 

320.  It  is  obvious,  however,  from  principles  already  explained, 
that  the  power  will  in  this  case  move  over  5040  times  as  great 
a  space  as  the  weight.    It  is  on  this  principle  that  the  screw  affords 
the  means  of  measuring  very  minute  spaces,  and  hence  is  derived 
the  Micrometer  Screw.     The  very  slow  motion  which  may  be  im- 
parted to  the  end  of  a  screw,  while  the  power  moves  over  a 
space  vastly  greater,  renders  if  peculiarly  adapted  to  this  pur- 
pose.    For  example,  suppose  a  screw  to  be  so  cut  as  to  have  50 
threads  in  an  inch  ;  then  each  revolution  of  the  screw  will  ad- 
vance its  point  through  the  50th  part  of  an  inch,  and  if  that  point 
acted  against  a  thread  or  wire,  it  would  move  it  over  a  gradua 
ted  space  only  that  distance  in  a  whole  revolution  of  the  screw. 
Now  suppose  the  head  of  the  screw  to  be  a  circle  an  inch  in  di- 
ameter, and  of  course  something  more  than  three  inches  in  cir- 
cumference.    This  circumference  may  easily  be  divided  into  a 
hundred  equal  parts,  distinctly  visible  ;  and  if  a  fixed  index  be 
applied  to  it,  the  hundredth  part  of  a  revolution  of  the  screw 
may  be  observed,  by  noting  the  passage  of  one  division  of  the 
head  under  the  index.     But  the  hundredth  part  of  a  revolution 
carries  the  point  of  the  screw  only  through  the   (y^  of  sV— ) 
aoV^th  part  of  an  inch.     Such  an  apparatus  is  frequently  at- 
tached to  the  limbs  of  graduated  instruments,  for  the  purposes 
of  astronomical  and  other  observations  ;  by  which  means,  a  por- 
tion of  the  graduated  arc  no  greater  than  the  \  Oth  part  of  a  sec- 
ond, can  be  estimated. 

In  like  manner,  any  other  small  space  may  be  measured  by 
the  aid  of  the  Micrometer  Screw.  Thus,  any  aliquot  part  of  a 
pound,  or  an  ounce,  in  the  steelyards,  may  be  found  by  adapting 
the  screw  to  the  counterpoise  so  as  to  move  it  slowly  over  the 
space  between  two  notches,  and  at  the  same  time  point  out.  by 
an  index  on  its  head,  the  exact  portion  of  the  space  over  which 
it  passes. 

THE   WEDGE. 

321.  If  instead  of  moving  a  load  on  an  inclined  plane,  the 
plane  itself  is  moved  beneath  the  load,  it  then  becomes  a  wedge. 
Thus,  if  a  perpendicular  beam  have  one  end  resting  upon  an 
inclined  plane,  (the  beam  being  so  secured  as  to  be  capable  of 
moving  only  up  and  down,)  and  the  plane  be  drawn  under  it,  the 
beam  will  be  elevated ;  and  the  power  required  to  effect  this 


MECHANICS.  227 

will  be  to  that  required  to  raise  the  beam  when  applied  directly 
to  it,  as  the  height  of  the  plane  to  its  length  : — or,  considering  the 
plane  as  a  half  wedge,  the  proportion  will  be,  as  half  the  back  of 
the  wedge  to  its  length.  (Art.  146.) 

322.  In  the  arts  and  manufactures,  wedges  are  used  where  an 
enormous  force  is  to  be  exerted  through  a  very  small  space. 
Thus  it  is  resorted  to  for  splitting  masses  of  timber  or  stone. 
Ships  are  raised  in  docks  by  wedges  driven  under  their  keels. 
The  wedge  is  the   principal  agent  in  the  oil-mill.     The  seeds 
from  which  the  oil  is  to  be  extracted  are  introduced  into  hair 
bags,  and  placed  between  planes  of  hard  wood.     Wedges  in- 
serted between  the  bags  are  driven  by  allowing  heavy  beams  to 
fall  on  them.     The  pressure  thus  excited  is  so  intense,  that  the 
seeds  in  the  bags  are  formed  into  a  mass  nearly  as  solid  as  wood. 
Instances  have  occurred  in  which  the  wedge  has  been  used  to 
restore  a  tottering  edifice  to  its  perpendicular  position.     All  cut- 
ting and  piercing  instruments,  such  as  knives,  razors,  scissors, 
chisels,  nails,  pins,  needles,  awls,  &c.  are  wedges.     The  angle 
of  the  wedge  in  these  cases,  is  more  or  less  acute,  according  to 
the  purpose  to  which  it  is  applied.     In  determining  this,  two 
things  are  to  be  considered — the  mechanical  power,  which  is  in- 
creased by  diminishing  the  angle  of  the  wedge,  (Art.  146,)  and 
the  strength  of  the  tool,  which  is  always  diminished  by  the  same 
cause.     There  is,  therefore,  a  practical  limit  to  the  increase  of 
the  power,  and  that  degree  of  sharpness  only  is  to  be  given  to 
the  tool,  which  is  consistent  with  the  strength  requisite  for  the 
purpose  to  which  it  is  to  be  applied.     In  tools  intended  for  cut- 
ting wood  the  angle  is  generally  about  30°  ;  for  iron  it  is  from 
50°  to  60°  ;  and  for  brass,  from  80°  to  90°.     Tools  which  act  by 
pressure  may  be  made  more  acute  than  those  which  are  driven 
by  a  blow  ;  and,  in  general,  the  softer  and  more  yielding  the 
substance  to  be  divided  is,  and  the  less  the  power  required  to  act 
upon  it,  the  more  acute  the  wedge  may  be  constructed.* 

323.  In  many  cases,  the  utility  of  the  wedge  depends  on  that 
which  is  entirely  omitted  in  the  theory,  viz.  the  friction  which 
arises  between  its  surface  and  the  substance  which  it  divides. 
This  is  the  case  when  pins,  bolts,  or  nails,  are  used  for  binding 
the  parts  of  structures  together  ;  in  which  case  were  it  not  for 
the  friction,  they  would  recoil  from  their  places,  and  fail  to  pro- 
duce the  desired  effect.     Even  when  the  wedge  is  used  as  a 
mechanical  engine,  the  presence  of  friction  is  absolutely  indis- 
pensable to  its  practical  utility.     The  power  generally  acts  by 
successive  blows,  and  is  therefore  subject  to  constant  intermission, 
and  but  for  the  friction,  the  wedge  would  recoil  between  the 

*  Lardner. 


228  NATURAL   PHILOSOPHY. 

intervals  of  the  blows  with  as  much  force  as  it  had  been  driven 
forward,  and  the  object  of  the  labor  would  be  continually  frus- 
trated. 

GENERAL  REMARKS  ON  MACHINERY. 

324.  Archimedes-  is  said  to  have  boasted  to  King  Hiero,  that 
"if  he  would  give  him  a  place  to  fix  his  machine,  (a  *ou  tfrw,)  he 
would  move  the  world."     Yet  there  can  be  no  machine  by  the 
aid  of  which  Archimedes  could  move  the  world,  in  any  other 
way,  than  by  moving,  himself,  over  as  much  more  space  than 
that  over  which  he  moved  the  earth,  as  his  weight  was  less  than 
that  of  the  whole  earth.     If  Archimedes  had  received  the  place 
he  desired,  and  had  also  employed,  what  was  equally  desirable, 
a  machine  which  operated  free  of  all  resistance,  he  must  have 
moved  with  the  velocity  of  a  cannon  ball,  to  have  shifted  the 
earth  only  the  27  millionth  part  of  an  inch  in  a  million  of  years.* 

325.  Machines  are  divided  into  two  classes,  those  intended 
simply  to  sustain  a  weight  and  those  intended  to  move  it.     In 
machines  of  the  first  class,  estimating  the  effect  by  the  weight 
sustained,  it  is  evident  that  the  efficacy  of  the  power  is  increased. 
By  means  of  a  lever,  for  example,  a  man  may  sustain  a  weight 
ten  times  as  great  as  he  could  by  his  unaided  strength.     We  may 
perceive,  however,  on  closer  examination,  that  he  does  not  in  fact 
bear  the  whole  weight,  but  only  one  tenth  part  of  it.     Let  it  be 
a  lever  of  the  second  kind,  where  the  weight  is  ten  times  nearer 
the  fulcrum  than  the  end  is  to  which  the  power  is  applied.     Now 
the  hand  that  supports  this  end  performs  the  same  office  as  the 
second  fulcrum  in  a  lever  of  the  first  kind  ;  and  since  (Art.  102) 
the  pressure  on  each  fulcrum  is  inversely  as  its  distance  from  the 
weight,  therefore,  in  the  present  case,  nine  parts  out  of  the  ten 
are  borne  by  the  prop,  and  only  one  by  the  power.     If  (says  Car- 
not)  Archimedes  had  obtained  his  "  fixed  point,"  it  would  not, 
in  reality,  have  been  Archimedes,  but  the  fixed  point  that  would 
have  sustained  the  earth. 

In  machines  of  the  second  class,  the  effect  is  not  estimated 
simply  by  the  weight  moved,  but  we  must  take  into  the  account 
the  time  occupied  in  moving  it  a  certain  distance,  that  is,  the 
velocity.  Hence,  the  effect  of  moving  powers  is  estimated  by 
the  product  of  the  weight  moved  multiplied  by  the  velocity,  or 
it  is  measured  by  the  momentum  produced.  Moreover,  in  the 
former  case,  all  resistance  from  friction,  the  rigidity  of  ropes,  and 
so  on,  conspire  with  the  power  in  sustaining  the  weight ;  but  in 
machines  of  the  second  class,  all  such  resistances  oppose  the  ac- 
tion of  the  power,  and  require  a  greater  power  for  a  given  weight 

*  Ed.  Encyc.,  XII,  578. 


MECHANICS.  229 

than  would  be  necessary,  if  the  power  were  applied  directly  to 
the  weight.* 

326.  Hence  it  will  be  inferred,  that  no  momentum,  or  effective 
force,  is  gained  by  any  of  the  mechanical  powers,  or  by  any  ma- 
chine.    If  a  man,  with  his  naked  hands,  can  lift  to  a  given 
height,  as  one  foot,  only  150  pounds  in  one  second,  it  is  impossi- 
ble for  him  to  perform  any  more  labor  than  this  by  any  me- 
chanical contrivances.  •[     On  the  contrary,  when  the  structure  of 
the  machine  is  complicated,  there  is  a  loss  of  force,  by  employing 
the  machine  instead  of  the  naked  hands,  proportioned  to  the  re- 
sistance of  the  parts  of  the  machine  itself.     It  is  to  be  remarked, 
however,  that  this  doctrine  proceeds  on  the  supposition  that  the 
useful  effect  produced  is  estimated  from  the  joint  product  of  the 
force,    velocity,   and    time.      Thus,  FxTxV=iFx2TxV=2Fx 
iTxV=FxiTx2V,  &c.J     A  convenient  method  of  estimating 
different  forces  is  to  draw  a  heavy  weight  out  of  a  well,  by  a 
rope  passing  horizontally  over  a  fixed  pulley,  near  the  top  of  the 
well.     Suppose  that  a  man  can  draw  up  a  rock  weighing   100 
pounds,  through  the  space  of  50  feet  in  one  minute.     He  would, 
of  course,  be  able  to  draw  up  ten  such  masses  in  ten  minutes, 
weighing  in  all   1000  pounds.     Now  by  passing  the  rope  over 
five  pulleys,  (allowing  nothing  for  the  friction  of  the  pulleys,)  he 
might  with  the  same  force  lift  the  whole  1000  pounds  at  once  ; 
but  it  would  rise  ten  times  as  slowly  as  the  100  pounds  did  be- 
fore, and  consequently  would  be  ten  minutes  in  reaching  the  top. 
Therefore,  in  a  given  time,  it  appears  that  the  man  would  raise 
the  same  weight  through  a  given  space,  with  or  without  the  aid 
of  machinery.     In  the  former  case,  the  100  pounds  might  have 
been  raised  during  the  ten  minutes  through  the  space  of  500  in- 
stead of  50  feet ;  but   100x500x10=1000x50x10:  so  that  the 
labor  performed  would  have  been  the  same  in  both  cases.     Let 
us  suppose  that  P  is  a  power  amounting  to  an  ounce,  and  that 
W  is  a  weight  amounting  to  50  ounces,  and  that  P  elevates  W 
by  means  of  a  machine.     In  virtue  of  the  property  already 
stated,  it  follows,  that  while  P  moves  through  50  feet,  W  will  be 
moved  through  1  foot ;  but  in  moving  P  through  50  feet,  fifty 
distinct  efforts  are  made,  by  each  of  which,  if  applied  directly,  1 
ounce  would  be  moved  through  1  foot.§ 

327.  What  then,  it  may  be  asked,  are  the  advantages  gained 
by  Machinery  ?    The  advantages  still  are  very  great,  for  the  fol- 
lowing reasons. 

(1.)  By  the  aid  of  machinery,  we  can  frequently  apply  our  force 
to  much  better  purpose.     Thus  in  lifting  a  weight  out  of  a  well, 

*  Venturoli's  Mechanics,  p.  164. 

+  Emerson's  Mechanics,  p.  1501     Cavallo,  Nat.  Phil.  I,  251. 

I  Gregory,  I,  343.  §  Lardner's  Mechanics,  p.  162. 


230  NATURAL   PHILOSOPHY. 

or  in  raising  ore  oijt  of  a  mine,  it  is  obvious  with  how  much 
more  effect  a  man  can  work  at  the  arm  of  a  windlass,  than  he 
could  draw  directly  upon  the  rope  by  stooping  over  the  well. 
So  in  raising  a  rock  from  its  bed  by  means  of  a  handspike  or 
crowbar,  we  can  easily  see  how  much  more  effectually  we  can 
bring  our  force  to  bear  upon  it  than  we  could  do  with  our  naked 
hands. 

(2.)  By  the  aid  of  machinery,  a  man  may  be  able  to  perform 
works  to  which  his  naked  strength  would  be  wholly  incompetent. 
Thus,  as  in  the  preceding  example,  one  might  be  able  to  lift  a 
rock  from  its  bed  with  a  handspike,  upon  which  he  could  make 
no  impression  with  his  naked  hands :  or,  by  means  of  pulleys, 
he  might  raise  a  box  of  merchandise  from  the  hold  of  a  ship, 
which  he  could  not  start  at  all  with  his  unassisted  force.  In 
each  of  these  cases,  if  the  weight  could  be  divided  into  small  parcels, 
and  if  the  force  could  be  as .  advantageously  applied  without 
machinery  as  with  it,  the  labor  would  be  performed  as  easily  in 
a  given  time  in  one  way  as  in  the  other.  But  it  might  not  be 
possible  or  at  least  convenient  thus  to  divide  it.  Or  if,  instead 
of  dividing  it  into  a  number  of  parcels,  the  same  number  of  men 
could  act  directly  upon  the  weight  at  once,  the  amount  of  labor 
which  they  would  all  exert  in  raising  the  weight  without  ma- 
chinery, would  be  the  same  as  that  which  the  single  man  before 
supposed  would  exert  with  his  machinery.  But  it  might  not  be 
convenient  to  assemble  so  many  hands  at  a  time  ;  or  perhaps 
such  a  number  could  not  work  advantageously  together.  A 
farmer  has  many  occasions  for  lifting  or  removing  great  weights 
when  his  laborers  are  not  more  in  number  than  two  or  three  in 
all.  These  must  therefore  perform  the  labor  of  50  times  as 
many  men  by  being  50  times  as  long  about  it.  Thus,  in  the  ex- 
ample given  on  page  118,  of  a  combination  of  the  mechanical 
powers  employed  to  haul  a  ship  on  the  stocks,  where  a  single 
man  turning  on  a  winch,  with  the  force  of  100  pounds  exerts  a 
force  on  the  ship  amounting  to  161 5  tons,  the  ship  would  move 
as  much  slower  than  the  hand  as  100  pounds  is  less  than  16l£ 
tons  ;  and  consequently  a  great  length  of  time  would  be  required 
for  an  individual  to  perform  this  labor,  even  supposing  no  resist- 
ance to  be  encountered  from  the  machinery  itself. 

(3.)  Machinery  frequently  enables  a  man  to  exert  his  whole 
force  in  circumstances  where,  without  such  aid,  he  could  employ 
but  a  part  of  it.  Thus,  in  winding  silk  or  thread,  to  turn  a  single 
reel  might  not  require  one  fiftieth  part  of  the  force  which  the 
laborer  is  capable  of  exerting.  Suitable  machinery  would  enable 
him  to  turn  fifty  spools  at  once. 

(4.)  But  the  most  striking  advantage  of  machinery,  is  not 
found  in  the  facilities  which  it  lends  to  the  personal  strength  of 
man.  It  lies  in  this,  that  it  affords  the  means  of  calling  in  to 
his  assistance  the  superior  powers  of  the  horse  and  the  ox,  of 


MECHANICS.  231 

water,  of  wind,  and  especially  of  steam.  Here  we  find  the  ex- 
cellence of  mechanical  contrivances  fully  exhibited ;  and  no- 
where else  has  the  inventive  genius  of  man  displayed  itself  to 
so  great  advantage.  But  here,  as  in  all  other  cases,  the  various 
combinations  of  mechanical  powers  produce  no  force  :  they  only 
apply  it.  They  form  the  communication  between  the  moving 
power  and  the  body  moved ,  and  while  the  power  itself  may  be 
incapable  of  acting  except  in  one  direction,  we  are  able  by 
means  of  cranks,  levers,  and  toothed  wheels,  to  direct  and  modify 
that  force  to  suit  our  convenience  or  necessities.  Every  one 
may  see  examples  of  this  in  the  construction  of  the  most  com- 
mon saw-mill  or  flour-mill,  turned  by  water.  In  a  mill  for  grind- 
ing wheat,  the  stones  are  required  to  move  horizontally,  while 
the  action  of  the  waterfall  is  perpendicular.  We  therefore  re- 
ceive the  whole  force  on  the  circumference  of  a  wheel,  and 
transmit  it  through  several  intermediate  wheels  to  the  revolving 
stone,  where  the  grinding  is  performed.  So  in  a  saw-mill,  the 
water  first  communicates  a  rotary  motion  to  the  wheel,  and  this 
motion  is  converted  by  means  of  a  crank  into  what  is  called  a 
reciprocating  motion,  as  that  of  the  saw  in  its  ascent  and  de- 
scent. By  means  of  wheel- work  the  velocity  of  the  moving  body 
is  increased  or  diminished  at  pleasure. 

328.  In  short,  machines  enable  us  to  form  a  convenient  com- 
munication between  the  power  and  the  weight ;  to  give  to  the 
weight  any  required  direction  or  velocity  ;  to  apply  force  to  the 
best  advantage  ;  to  vary  the  circumstances  of  velocity  and  time 
as  the  amount  of  our  force  may  require  ;  and  to  bring  to  our  aid 
the  great  moving  powers  that  exist  in  nature.  Our  next  object, 
therefore,  will  be  to  see  by  what  particular  methods  these  several 
purposes  are  accomplished. 


CHAPTER  IV. 

REGULATION  OF  MACHINERY,  AND   CONTRIVANCES   FOR   MODI- 
FYING MOTION. 

329.  IT  is  highly  important  to  the  successful  operation  of  any 
machine,  that  its  motion  should  be  regular  and  uniform.  Jolts 
and  irregular  movements,  waste  the  power,  wear  upon  the  ma- 
chine, and  perform  the  work  unevenly.  The  sources  of  irregu- 


232 


NATURAL   PHILOSOPHY. 


larity  are  various,  but  they  are  chiefly  the  three  following,  viz. 
variations  in  the  power,  variations  in  the  weight  or  resistance, 
and  changes  of  velocity  in  parts  of  the  machine  itself.  Thus  in 
the  steam  engine,  the  fire  may  burn  with  more  or  less  intensity, 
and  produce  corresponding  quantities  of  the  moving  power ;  the 
load  to  be  carried  (as  that  of  a  steamboat)  may  be  much  greater 
at  one  time  than  at  another,  and  be  subject  to  sudden  changes ; 
and  the  motion  of  the  piston,  which  carries  the  machinery,  ceases 
altogether  at  the  highest  and  lowest  points,  and  would  move  a 
machine  by  hitches  or  separate  impulses,  were  there  no  contri- 
vance connected  with  it  for  keeping  up  a  uniform  motion. 

330.  The  kinds  of  apparatus  employed  to  obviate  these  diffi- 
culties, and  to  secure  uniform  movements  to  machines,  are,  in 
general,  called  REGULATORS.     Large  machines  or  engines  them- 
selves, in  consequence  of  their  inertia,  acquire  and  maintain  to  a 
considerable  extent,  uniformity  of  motion.     A  flour-mill  carried 
by  water,  when  it  has  acquired  a  certain  rate  of  going,  will  not 
suddenly  change  that  rate  by  any  alteration  in  the  force  of  the 
stream  ;  and  a  ship  sailing  between  the  opposite  forces,  arising 
from  the  impulse  of  the  wind  and  the  resistance  of  the  water, 
will  move  steadily  along,  notwithstanding  the  breeze  that  car- 
ries it  may  fluctuate  continually.     We  can  see  this  principle 
sometimes  operating  on  a  smaller  scale.     A  grindstone  turned 
by  a  winch  moves  steadily,  although  the'  force  applied  at  one 
part  of  the  revblution  is  much  greater  than  at  another.     Large 
grindstones  exhibit  the  advantage  of  this  principle  much  more 
than  small  ones.     But  in  many  instances,  this  natural  tenden- 
cy toward  uniform  motion  is  not  sufficient,  and  artificial  con- 
trivances are  introduced  expressly  for  this  purpose.     As  exam- 
ples of  regulators  we  may  especially  notice  three,  the  Pendulum, 
the  Fly  Wheel,  and  the  Governor. 

331.  The   Fly   Wheel  affords  the  Fig.  156. 
most  common  and  effectual  method 

of  equalizing  motion,  especially  in 
heavy  kinds  of  machinery.  It  con- 
sists of  a  heavy  wheel,  (Fig.  156,) 
affording  as  much  weight  as  possible 
under  as  small  a  surface,  in  order 
that  the  inertia  may  be  great,  while 
the  resistance  from  the  air  is  small. 
It  is  therefore  usually  a  heavy  hoop 
of  iron  with  thick  bars  of  the  same 
metal.  The  Fly  is  balanced  on  its 
axis,  and  so  connected  with  the  machinery  as  to  turn  rapidly 
around  with  it,  and  receiving  a  constant  impulse  from  the  mov- 
ing power,  it  becomes  a  magazine  or  repository  of  motion.  Con- 


MECHANICS.  233 

sequently,  by  its  inertia,  it  is  ready  to  supply  any  deficiency  of 
power  that  may  arise  from  the  sudden  diminution  of  the  moving 
force,  or  to  check  any  sudden  impulse  which  may  result  from  an 
accidental  excess  of  that  force.  Suppose,  for  example,  the 
handle  of  a  pump  to  be  connected  with  a  water  wheel,  and  to 
be  carried  by  it.  Here  the  power,  namely  the  waterfall,  is  con- 
stant, while  the  weight  is  subject  to  continual  alternations, 
amounting  to  a  heavy  load  as  the  piston  is  ascending,  but  op- 
posing scarcely  any  resistance  while  the  piston  is  descending 
The  motion,  therefore,  would  vary  between  nothing  and  a  highly 
accelerated  velocity,  and  the  machinery  would  be  subject  to 
constant  strains  and  jolts.  A  Fly  prevents  these  alternations, 
and  renders  the  ascent  and  descent  of  the  piston  nearly  uniform. 
In  pile  engines  or  stamping-mills  a  team  of  horses  is  sometimes 
employed  to  raise  a  heavy  weight,  which,  when  at  a  certain  eleva- 
tion, is  suddenly  disengaged,  and  falls  with  great  force.  As  the  dis- 
engagement is  instantaneous,  the  horses  would  instantly  tumble 
down  were  not  their  motion  checked  by  some  contrivance  which 
should  prevent  the  machinery  from  receiving  any  sudden  in- 
crease of  velocity.  This  purpose  is  completely  answered  by  the 
Fly.* 

332.  Besides  the  use  of  the  Fly  Wheel  in  regulating  the  action 
of  machinery,  it  is  employed  for  the  purpose  of  accumulating 
successive  exertions  of  a  power,  so  as  to  produce  a  much  more 
forcible  effect  by  their  aggregation  than  could  possibly  be  done 
by  their  separate  actions.  If  a  small  force  is  repeatedly  applied 
in  giving  rotation  to  a  Fly  Wheel,  and  is  continued  until  the 
wheel  has  acquired  a  very  considerable  velocity,  such  a  quantity 
of  force  will  be  at  length  accumulated  in  its  circumference,  as  to 
overcome  resistance  and  produce  effects  utterly  disproportionate 
to  the  immediate  action  of  the  original  force.  Thus  it  would  be 
very  easy  in  a  few  seconds,  by  the  mere  action  of  a  man's  arm, 
to  impart  to  the  circumference  of  a  Fly  Wheel,  a  force  which 
would  give  an  impulse  to  a  musket  ball  equal  to  that  which  it 
receives  from  a  full  charge  of  powder,  f 

The  same  principle  explains  the  force  with  which  a  stone  may 
be  projected  from  a  sling.  The  thong  is  swung  several  times 
around  by  the  arm  until  a  considerable  portion  of  force  is  ac- 
cumulated, and  then  it  is  projected  with  all  the  collected  force. 
If  a  heavy  leaden  ball  is  attached  to  the  end  of  a  strong  piece 
of  cane  or  whalebone,  it  may  easily  be  driven  through  a  board : 
by  taking  the  end  of  the  rod  remote  from  the  ball  in  the  hand, 
and  striking  the  board  a  smart  blow  with  the  end  bearing  the 
ball,  such  a  velocity  may  easily  be  given  to  the  ball  as  will  drive 
it  through  the  board.  J 


*  Gregory's  Mech.,  II,  14.  t  Library  of  Useful  Knowledge.  J  Jb. 

30 


234  NATURAL    PHILOSOPHY. 

333.  The  astonishing  effects  of  a  Fly  "Wheel,  as  an  accumu- 
lator of  force,  have  led  some  into  the  error  of  supposing  that 
such  an  apparatus  increases  the  actual  force  of  a  machine.  So  far 
from  this,  since  a  Fly  cannot  act  without  friction  and  resistance 
from  the  air,  a  portion  of  the  actual  moving  force  must  unavoid- 
ably be  lost  by  the  use  of  this  appendage.  In  cases,  however, 
where  a  Fly  is  properly  adjusted  and  applied,  this  loss  of  power 
is  inconsiderable,  compared  with  the  advantageous  distribution 
of  what  remains.  As  an  accumulator  of  force,  a  Fly  can  never 
have  more  force  than  has  been  applied  to  put  it  in  motion.  In 
this  respect  it  is  analogous  to  an  elastic  spring.  In  bending  a 
spring,  a  gradual  expenditure  of  power  is  necessary.  On  the 
recoil,  this  power  is  exerted  in  a  much  shorter  time  than  that 
consumed  in  its  production,  but  its  total  amount  is  not  altered. 
In  this  way  the  Fly  Wheel  is  used.  Thus,  in  mills  for  rolling 
metal,  the  water  wheel  or  other  moving  power,  is  allowed  for 
some  time  to  act  upon  the  Fly  alone,  no  load  being  placed  upon 
the  machine.  A  force  is  thus  gained  which  is  sufficient  to  roll  a 
large  piece  of  metal,  to  which,  without  such  means,  the  mill 
would  be  quite  inadequate.  In  the  same  manner,  a  force  may 
be  gained  by  the  arm  of  a  man  acting  on  a  Fly  for  a  few  seconds, 
sufficient  to  impress  an  image  on  a  piece  of  metal  by  an  instan- 
taneous stroke.  The  Fly  is,  there-  F-  157 
fore,  the  principal  agent  in  coining 
presses.  Its  power  is  often  trans- 
mitted to  the  working  point  by 
means  of  a  screw.  At  the  extrem- 
ities of  the  cross  arm  AB,  (Fig. 
157,)  which  works  the  screw,  two 


heavy  balls  of  metal  are  placed. 
When  the  arm  AB  is  whirled  r 


round, 

those  masses  of  metal  acquire  a 
momentum,  by  which  the  screw,  being  driven  for  ward,  urges  the 
die  with  immense  force  against  the  substance  destined  to  receive 
the  impression.  Some  engines  used  in  coining  have  Flies  with 
arms  four  feet  long,  bearing  one  hundredweight  at  each  of  their 
extremities.  By  turning  such  an  arm  at  the  rate  of  one  entire 
circumference  in  a  second,  the  die  will  be  driven  against  the 
metal  with  the  same  force  as  that  with  which  7500  pounds  weight 
would  fall  from  the  height  of  16  feet;  an  enormous  power,  if 
the  simplicity  and  compactness  of  the  machine  is  considered. 
fey  the  action  of  a  Fly,  working  in  this  manner,  is  produced  the 
open  work  of  fenders,  fire  grates,  and  sometimes  ornamental 
articles  wrought  in  metal.  The  cutting  tool,  shaped  according 
to  the  pattern  to  be  executed,  is  attached  to  the  end  of  a  screw ; 
and  the  metal  being  held  in  a  proper  position  beneath  it,  the  Fly 
is  made  to  urge  the  tool  downward  with  such  force  as  to  stamp 
out  pieces  of  the  required  figure.  When  the  pattern  is  compli- 


MECHANICS. 


cated,  and  it  is  necessary  to  preserve  with  exactness  the  relative 
situation  of  its  different  parts,  a  number  of  punches  are  impelled 
together,  so  as  to  strike  the  entire  piece  of  metal  at  the  same 
instant,  and  in  this  manner  the  most  elaborate  work  is  executed 
by  a  single  stroke  of  the  hand.* 

334.  To  maintain  a  uniform  ve-  Fig.  158. 

locity  with  a  varying  resistance,  one 
of  the  most  beautiful  contrivances 
ever  used  is  the  Governor,  (Fig. 
158,)  an  instrument  used  in  mill 
work,  but  the  application  of  which 
is  most  conspicuous  in  the  steam 
engine,  when  that  machine  is  ap- 
plied to  manufacturing  purposes. 
The  principle  on  which  the  efficacy 
of  this  instrument  depends,  is  easily 
explained.  Let  AB  be  a  vertical 
axis  which  is  made  to  revolve  by 
the  wheel  A,  acted  on  by  the  other  ( 
parts  of  the  machinery,  and  so  that 
it  always  revolves  with  a  velocity 
proportional  to  that  of  the  fly  wheel. 
Two  heavy  balls  C,  C  are  attached 
to  metal  rods,  which  work  on  a  pivot 
at  B,  so  that  they  are  capable  of 
receding  from  the  axis  AB.  As 
they  recede  from  the  axis,  the  joints  D,  D'  recede  from  one 
another,  and  the  joint  E  is  drawn  down.  This  joint  E  is  con- 
nected with  the  end  of  a  lever  or  a  system  of  levers,  the  action 
of  which  we  shall  presently  explain. 

Now  by  the  revolution  of  the  spindle  or  axis  AB,  the  balls  C, 
C  acquire  an  obvious  tendency  to  fly  off  from  the  axis,  and  this 
tendency  is  resisted  by  their  weight ;  so  that,  when  the  instru- 
ment is  revolving  with  a  certain  velocity,  the  balls  remain 
suspended,  and  neither  move  to  or  from  the  axis.  A  greater 
velocity,  by  giving  a  greater  centrifugal  force,  would  cause  the 
balls  to  fly  further  off,  (Art.  238,)  and  a  less  velocity  would 
cause  them  to  fall  toward  the  axis.  This  is  strictly  true  only 
when  the  range  of  balls  is  small,  compared  with  the  length  of 
the  rods  to  which  they  are  attached,  which,  however,  is  always 
the  case  in  practice.  If,  therefore,  the  action  of  the  levers  with 
which  the  joint  E  is  connected  is  directed  upon  the  first  mover 
in  such  a  manner,  that  its  energy  is  diminished  when  E  is  de- 
pressed, and  increased  when  E  is  elevated,  it  is  plain  that  the 
uniformity  of  velocity  which  is  sought  may  be  obtained.  Let 


*  Lardner. 


236  .  NATURAL   PHILC60PHY. 

us  suppose  that  the  levers  on  which  E  works  communicate  with 
a  valve  which  admits  steam  to  the  piston  of  a  steam  engine,  to 
which  this  Governor  is  applied  ;  and  suppose  that  when  E  is 
raised,  and  the  balls  C,  C  rest  in  their  seats,  the  valve  is  fully 
open,  so  as  to  allow  the  steam  to  flow  in  a  full  stream  to  the 
piston  ;  but  that,  according  as  E  is  depressed,  the  levers  gradu- 
ally close  the  valve,  so  as  to  admit  the  steam  in  a  constantly 
diminished  quantity.  Now  suppose  that  the  engine  has  been 
working  twenty  printing  presses,  and  that  the  action  of  ten  of 
them  is  suddenly  suspended.  The  engine  thus  loses  half  its 
load,  and  would,  if  the  same  power  of  steam  continued  to  be 
admitted,  move  with  about  twice  its  former  velocity.  But  the 
moment  an  increased  velocity  is  perceived  in  the  machine,  the 
balls  C,  C  recede  from  the  axis,  draw  down  the  joint  E,  partially 
close  the  valve,  and  check  the  supply  of  steam  to  the  cylinder. 
The  impelling  power  is  thus  diminished  ;  and  if  it  be  diminished 
in  exactly  the  same  degree  as  the  load,  the  machine  will  move 
with  its  former  velocity  ;  but  if  it  should,  at  first,  be  more  di- 
minished, the  velocity  will  be  less  than  its  former  velocity,  and 
the  balls  will  again  move  toward  the  axis  and  open  the  valve, 
and  will,  at  length,  settle  into  that  position  in  which  the  steam 
admitted  to  the  cylinder  is  exactly  proportioned  to  the  load  on 
the  machine  ;  and  the  proper  velocity  will  thus  be  restored.* 

CONTRIVANCES  FOR  MODIFYING  MOTION. 

335.  In  Chapter  III,  we  have  already  explained  the  mode  in 
which  motion  is  communicated,  and  its  velocity  regulated  by 
wheel-work.     We  proceed  now  to  consider  a  few  examples  of  the 
more  special  contrivances  by  which  motion  is  modified  to  suit 
particular  purposes,  recommending  it  to  the  student  of  mechanics 
to  make  himself  acquainted  with  other  contrivances  of  the  same 
nature,  by  the  actual  inspection  of  machinery  as  opportunity  mav 
ofier. 

336.  The  motion  required  for  a  particular  purpose  may  be  rec 
tilinear,  as  that  of  a  carriage  or  bucket  drawn  out  of  a  well,  or 
rotary,  as  in  ordinary  wheel-work,  or  reciprocating,  as  in  a  saw- 
mill or  a  pendulum. 

The  simplest  mode  of  producing  rectilinear  motion  is  by  means 
of  a  rope  or  chain,  instances  of  which  are  familiar  to  every  one. 
The  simplest  mode  of  changing  the  direction  is  by  means  of  pul- 
leys ;  but  toothed  wheels  are  also  extensively  employed  for  the 
same  purpose.f  The  connection  of  one  toothed  wheel  with 
another  is  called  gearing.  (Art.  294.)  When  both  wheels  with 


•  Library  of  Useful  Knowledge.  t  Emerson's  Mechanics,  Prop.  CX. 


MECHANICS. 


their  teeth  are  in  the  direction  of  the  same  plane,  it  is  called  spur 
gearing,  (Figs.  149,  150,  and  151  ;)  if  the  teeth,  instead  of  being 
cut  on  the  circumference  in  a  direction  parallel  to  the  axis,  are 
cut  obliquely,  so  that  if  continued  they  would  pass  round  the 
Fig.  159.  Fig.  160.  Fig.  161.  < 


axis  like  a  screw,  it  is  called  spiral  gearing,  (Fig.  1 59  ;)  and 
when  wheels  are  not  situated  in  the  same  or  parallel  planes, 
but  form  an  angle  with  each  other,  the  wheels  themselves  are 
sometimes  shaped  like  frustra  of  cones,  having  their  teeth  cut 
obliquely,  and  converging  toward  the  point  where  the  apex  of 
the  cone  would  be  situated,  and  it  is  then  called  bevel  gearing.* 
(Fig.  160.) 

337.  The  universal  joint  consists  of  two  shafts  or  arms,  each 
terminating  in  a  semicircle,  and  connected  together  by  means  of 
a  cross  upon  which  each  semicircle  is  hinged,  (Fig.  161.)  When 
one  shaft  is  turned,  either  to  the  right  or  left,  the  other  shaft 
turns  in  the  same  direction. 

The  ratchet  wheel,  (Fig.  162.)  is  used  to  prevent  motion  in  one 
direction  while  it  permits  it  in  the  opposite.  The  teeth  are  cut 
with  their  faces  inclining  as  in  the  figure,  and  a  catch  is  so  placed 
as  to  stop  the  wheel  in  one  direction,  while  it  slides  over  the 
teeth  without  obstruction  in  the  opposite  direction. 

Fig.  163. 


Fig.  162. 


338.  The  eccentric  wheel,  (Fig.  163,)  revolves  about  an  axis 
which  is  more  or  less  removed  from  the  center,  and,  consequent- 


*  Bigelow's  Tech.,  232. 


NATURAL   PHILOSOPHY. 


ly,  the  different  portions  of  the  circumference  move  with  differ- 
ent degrees  of  velocity.  Hence  if  this  wheel  is  made  to  act 
upon  a  shaft  or  pinion,  as  in  the  figure,  it  will  carry  it  with  a 
corresponding  movement.  In  orreries,  such  wheels  are  employed 
fcr  indicating  the  variable  velocities  of  the  heavenly  bodies,  as 
they  revolve  about  their  centers  of  motion. 

339.  RECIPROCATING  MOTION  is  produced  in  various  ways.     The 
most  common  method  is  by  means  of  the  crank.     In  Fig.  164,  a 
shaft  AB  is  urged  backward  or  forward,  (either  verti- 
cally or  horizontally,)  by  means  of  the  crank  ab,  mov-      ^g-^164- 
ing  on  a  wheel  H,  which  may  be  turned  by  water  or          •— i 
any  other  power  acting  at  H.     By  considering  the 
different  positions  of  the  crank  during  the  revolution 

of  the  wheel,  it  will  be  readily  seen  that  the  shaft 
will  move  up  and  down  like  the  saw  in  a  saw-mill, 
or  backward  and  forward,  a  use  to  which  it  is  applied 
in  polishing  plane  surfaces,  as  marble. 

The  motion  produced  by  cranks  is  easy  and  gradu- 
al, being  most  rapid  in  the  middle  of  the  stroke,  and 
gradually  retarded  toward  the  extremes  ;  so  that 
shocks  and  jolts  in  the  moving  machinery  are  dimin- 
ished, or  wholly  prevented  by  their  use.* 

The  steam-engine,  as  seen  in  steamboats,  furnishes 
to  the  student  of  mechanics  a  valuable  opportunity 
of  observing  various  contrivances  for  producing,  regulating,  and 
modifying  motion.  Levers  and  wheels  of  various  kinds  and  va- 
riously connected  ;  fly  wheels  and  cranks  ;  circular  and  recipro- 
cating motions  ;  and  numerous  other  particulars  which  appertain 
to  the  "  elements  of  machinery,"  are  there  seen  to  the  greatest 
advantage. 

340.  The  arch  head,  (Fig.  165,)  is  a  circular     _  Fig- 165. 
form  given  to  the  end  of  a  lever,  that  moves  on 

a  gudgeon  or  pivot  in  a  vertical  plane,  (like  the 
working  beam  of  a  steam-engine,)  by  means  of 
which  the  piston,  or  weight,  whatever  it  be,  is 
made  to  ascend  and  descend  perpendicularly  in 
a  straight  line,  with  a  uniform  motion,  while  the 
end  of  the  lever  itself  works  in  the  arc  of  a  cir- 
cle, and  while  the  power,  if  fixed  in  the  usual 
way  at  one  point  at  the  extremity  of  the  lever, 
would  lose  a  part  of  its  efficacy,  as  its  direction 
became  oblique  to  that  of  the  lever.  In  the 
figure,  BD  represents  the  arch  head  of  a  working  beam,  which 
turns  on  a  gudgeon  at  C.  At  the  top  B  is  applied  a  flexible 
chain  resting  loosely  upon  the  arch,  and  always  maintaining  a 


*  Bigelow. 


MECHANICS. 


239 


Fig.  166. 


vertical  position  in  every  situation  of  the  beam ;  and,  being  a 
tangent  to  the  circle  at  the  point  of  contact,  always  acting  at 
right  angles  to  the  lever,  and  consequently  imparting  to  the 
weight  a  uniform  motion.  In  cases  where  force  is  to  be  exerted 
only  in  one  direction,  as  in  working  a  pump,  the  power  is  ap- 
plied while  the  piston  is  descending  and  drags  up  the  other  end 
of  the  lever.  Here,  in  the  descent  of  the  working  end,  so  small 
a  resistance  is  encountered,  that  it  is  sufficient  to  give  a  slight 
preponderance  to  that  end,  and  it  will  drag  up  the  piston  after 
the  moving  force  is  withdrawn  which  carried  it  down.  But  in 
cases  where  a  continued  force  is  required,  both  in  the  ascent  and 
descent  of  the  piston,  (as  in  most  kinds  of  manufacturing  opera- 
tions,) then  the  flexible  chain,  being  incapable  of  forcing  the 
weight  upward,  is  inapplicable,  but  the  object  is  attained  by 
arming  the  arch  head  with  teeth  that  act  on  rack  work  in  the 
upper  end  of  the  piston  rod. 

341.  The  knee  joint,  (Fig.  166,) 
is  a  contrivance  by  which  a  con- 
stantly increasing  resistance  is 
overcome  by  a  force  which  acts 
nearly  uniformly.  In  the  figure, 
we  have  a  representation  of  it  as 
exemplified  in  printing  presses. 
Here,  when  the  platen  (P)  first 
descends  upon  the  sheet,  the  re- 
sistance is  very  slight,  but  as  the 
lever  by  which  the  press  is  worked 
is  pulled  still  further,  the  resist- 
ance rapidly  increases,  until,  with- 
out mechanical  aid,  it  would  re- 
quire a  laborious  and  exhausting 
effort,  to  render  the  contact  suffi- 
ciently close  to  give  a  perfect  im- 
pression. This  difficulty  is  com- 
pletely obviated  by  means  of  a 
combination  of  levers  resembling 
the  knee  joint,  where  the  efficacy 
of  the  power  is  continually  increased  as  the  levers  approach 
nearer  to  the  same  straight  line  ;  so  that  without  any  additional 
effort  on  the  part  of  the  workman,  the  pressure  is  augmented  a 
thousand  fold.  The  action  of  this  joint  is  exemplified  in  opening 
a  pair  of  compasses,  or  a  Gunter's  scale  which  opens  on  a  joint 
at  the  center.  At  first,  the  thrust  exerted  by  the  ends  is  slight ; 
but  as  the  two  parts  approach  the  direction  of  a  straight  line 
with  each  other,  the  thrust  rapidly  increases,  until  it  becomes 
immensely  great.* 

*  See  an  able  article  on  this  subject  by  Prof.  Fisher,  American  Journal  ef  Science, 
Vol.  Ill,  p.  311. 


240  NATURAL   PHILOSOPHY. 

We  shall  conclude  this  chapter  with  a  few  practical  rules  for 
the  construction  and  management  of  machinery,  selected  chiefly 
from  Emerson's  Mechanics. 

342.  If  the  given  power  is  not  able  to  overcome  the  given  re 
sistance  when  directly  applied,  then  we  must  employ  a  longer 
time  by  making  the  power  move  over  a  greater  space  than  the 
weight,  in  order  to  make  the  momentum  of  the  power  exceed 
that  of  the  weight.     This  is  effected  by  means  of  machinery 
We  must  consider  that  the  power  must  have  a  momentum  which 
is  equal  to  the  amount  of  all  the  resistances  to  be  overcome,  in 
order  just  to  sustain  or  counterbalance  the  weight ;  and  to  put 
the  machine  in  motion,  a  greater  power  must  be  allowed,  more 
or  less,  according  to  the  velocity  with  which  the  machine  is  re- 
quired to  move.     The  ratio  between  the  power  and  the  weight 
in  a  given  machine,  when  in  equilibrium,  may  be  ascertained  by 
observing  the  comparative  spaces  over  which  they  move  in  a  given 
time  ;  the  power  being  as  much  less  than  the  weight,  as  its  space 
is  greater.     A  due  proportion  between  the  two  must  also  be  ob- 
served ;  for  if  the  machine  or  engine  is  competent  to  overcome 
the  resistance,  and  perform  its  work  in  a  convenient  time,  it  is 
sufficient  for  the  end  proposed  ;  and  to  increase  the  power  any 
further,  must  not  only  be  a  needless  expense,  but  the  engine 
would  lose  time  in  working.     If  a  weight  is  to  be  moved  but  a 
little  way,  the  lever  is  the  most  simple,  easy,  and  ready  instru- 
ment.    If  the  weight  be  very  great,  the  common  screw  is  prefer- 
able.    But  if  the  weight  is  to  be  moved  over  a  great  space,  the 
wheel  and  axle,  the  pulley,  or  a  system  of  pulleys,  or  the  endless 
screw,  (Fig.  97,)  is  to  be  employed.     When  great  wheels  are 
wrought  by  men  or  cattle,  their  axes  are  most  advantageously 
placed  perpendicularly,  as  in  Figure  154  ;  if  wrought  by  water, 
'they  are  placed  horizontally. 

343.  But  most  machines  are  combinations  of  some  or  all  of 
the  mechanical  powers.     Thus  the  lever  is  combined  with  the 
screw  in  a  common  press ;  the  wheel  and  axle  with  pulleys  in 
various  ways,  and  with  the  endless  screw ;  pulleys  are  combined 
with  pulleys,  and  wheels  with  wheels.     The  wedge  is  the  only 
one  among  the  mechanical  powers,  that  does  not  admit  of  com- 
bination with  others.     In  wheels  with  teeth,  the  number  of  teeth 
that  play  together  in  two  wheels  ought  to  be  prime  to  each  other, 
that  the  same  teeth  may  not  meet  at  every  revolution,  but  as 
seldom  as  possible.     The  strength  of  every  part  of  a  machine 
ought  to  be  made  proportional  to  the  stress  it  is  to  bear ;  and  no 
part  must  be  stronger  or  heavier  than  is  necessary,  for  all  super- 
fluous matter  is  nothing  but  a  dead  weight  upon  the  machine, 
and  serves  for  nothing  but  to  clog  its  motion.     The  accomplished 
mechanic  contrives  all  the  parts  to  last  equally  well,  so  that 
vvhen  the  machine  fails,  every  part  shall  be  worn  out. 


MECHANICS.  241 

344.  Every  machine  ought  to  be  made  of  as  few  parts,  and  as 
simple  as  possible,  to  answer  its  purpose  ;  not  only  because  the 
expense  of  making  and  repairing  will  be  less,  -but  it  will  be  less 
liable  to  get  out  of  order.  Any  useless  motions  also  waste  some 
portion  of  the  power.  Uniformity  or  steadiness  of  motion  is  care- 
fully to  be  preserved.  All  these  advantages  are  more  easily  at- 
tained in  large  than  in  small  machines.  All  mechanical  errors 
have  a  less  ratio  to  the  motion  of  the  machine  in  great  machines 
than  in  small  ones,  and  these  will  therefore  work  with  more  uni- 
formity and  exactness,  although,  being  proportionally  weaker, 
they  are  less  able  to  resist  any  violent  shocks.* 


CHAPTER   V. 

OF  FRICTION. 

345.  THE  term  Friction,  in  its  usual  acceptation,  being  gener- 
ally understood,  we  have  already  employed  it  in  the  foregoing 
pages,  but  we  proceed  now  to  inquire  more  particularly  re- 
specting its  nature,  the  laws  of  its  action,  and  its  effects  upon 
machines. 

In  investigating  the  mathematical  principles  of  Mechanics,  we 
first  proceed  on  the  supposition  that  the  forces  in  question  act 
without  any  impediments ;  that  the  surfaces  which  move  in  con- 
tact are  perfectly  polished  and  suffer  no  friction ;  that  axes  and 
pivots  are  mathematical  lines  and  points ;  that  ropes  are  perfectly 
flexible  ;  and,  in  short,  that  the  power  is  transmitted  through  the 
machine  to  the  working  point  without  sustaining  the  least  loss  or 
diminution.  Great  simplicity  is  attained  by  first  bringing  the 
subject  to  this  ideal  standard  of  perfection,  and  afterward  making 
suitable  allowances  for  all  those  causes  which  operate  in  any 
given  case  to  prevent  the  perfect  action  of  a  machine. 

346.  Surfaces  meet  with  a  certain  degree  of  resistance  in 
moving  on  each  other,  in  consequence  of  the  mutual  cohesion  of 
the  parts,  a  principle  which  has  the  greater  influence  in  any 
given  case  in  proportion  as  the  surfaces  are  smooth.     But  a 
much  greater  resistance  arises  from  the  asperities  which  the 
surfaces  of  all  bodies  have,  though  in  very  different  degrees, 
according  to  their  different  degrees  of  smoothness.     An  extreme 
case  is  that  of  two  brushes  moving  on  each  other,  the  hairs  of 
which  become  interlaced,  (especially  when  the   brushes  are 

*  Emersou  s  Mechanics,  4to.  p.  175. 
31 


242  NATURAL   PHILOSOPHY. 

pressed  together,)  and  oppose  a  great  resistance.  Even  bodies 
apparently  very  smooth,  as  polished  metals,  exhibit  under  the 
microscope  numerous  inequalities.  Under  the  solar  microscope, 
the  finest  needle  exhibits  a  surface  as  rough  as  the  coarsest  iron 
tools  do"  when  viewed  by  the  naked  eye.  To  these  inequalities 
of  surface,  is  principally  ascribed  the  friction  of  bodies,  when 
closely  in  contact ;  the.  prominent  parts  interlock  with  one 
another,  or  meet,  and  must  be  broken  down  before  the  surfaces 
can  move.  Hence,  friction  is  diminished  by  processes  which 
level  these  inequalities,  either  by  polishing  the  surface,  or  by 
coating  it  with  some  lubricating  substance  which  fills  up  the 
cavities. 

347.  Forces  of  this  nature,  which  act  by  the  resistance  they 
occasion  to  motion,  are  called  passive  forces.     They  produce  very 
different  effects  in  machines  when  in  a  state  of  equilibrium,  and 
in  a  state,  of  motion.  '  In  the  one  case  they  assist  the  power ;  in 
the  other  case  they  oppose  it.     Thus,  a  weight  placed  on  an  in- 
clined plane,  will  require  a  less  power  to  support  it  in  consequence 
of  the  friction  of  the  plane  ;  and  a  weight  suspended  by  a  rope 
passing  over  a  pulley  will  require  a  less  weight  to  balance  it,  on 
account  of  the  friction  of  the  axle.     But  the  same  passive  forces 
operate  in  just  the  contrary  way  when  a  machine  is  to  be  put  in 
motion ;  for  then  a  power  must  be  applied,  which  is  sufficient 
not  only  to  overcome  the  weight  itself,  but  also  the  amount  of  all 
the  resistances.     For  example,  in  order  to  draw  a  load  up  an 
inclined  plane,  we  have  to  overcome  not  only  the  force  of  gravity 
by  which  the  load  endeavors  to  descend  down  the  plane,  but 
also  the  amount  of  the  friction  and  all  the  other  resistances  which 
impede  its  motion,  although  the  load  would  be  kept  from  descend- 
ing, that  is,  in  a  state  of  equilibrium,  by  a  less  force  in  conse- 
quence of  these  resistances.     The  principle  is  most  strikingly 
observed  in  the  wedge,  where  the  difficulty  of  making  the  wedge 
advance,  is  greatly  increased  by  friction,  but  the  same  cause 
operates  to  prevent  it  from  recoiling. 

348.  Two  philosophers  /->f  great  eminence  have  severally  per- 
formed an  extensive  series  of  experiments  on  friction,  namely, 
M.  Coulomb,  member  of  the  Academy  of  Sciences  at  Paris,  and 
Professor  Vince,  of  the  University  of  Cambridge  in  England 
and  upon  their  investigations  is  founded  a  great  part  of  all  that 
is  known  with  precision  respecting  the  laws  of  friction. 

The  forms  under  which  this  sort  of  resistance  presents  itself, 
are  chiefly  of  two  kinds,  namely,  that  of  bodies  sliding,  and  of 
bodies  rolling  on  each  other.  To  the  former  of  these  let  us  first 
attend.  Experiments  on  the  friction  of  sliding  bodies  may  be 
made,  either  by  placing  them  on  a  table,  and  observing  the 
weights  which  they  respectively  require  to  drag  them  along  the 


MECHANICS.  243 

wble,  or  by  placing  tli&m  on  an  inclined  plane,  and  observing  at 
what  angle  the  plane  must  be  elevated  in  order  that  the  body 
may  begin  to  slide.  In  the  former  case,  the  table  is  prepared  by 
attaching  a  vertical  pulley  to  one  edge  over  which  a  string  is 
passed,  one  end  being  connected  to  the  body  in  question,  and  the 
other  end  to  a  pan,  like  that  of  a  balance,  for  containing  weights. 
From  this  simple  arrangement,  a  great  variety  of  particulars  may 
be  ascertained  respecting  the  friction  of  sliding  surfaces.  A  body 
shaped  like  a  brick,  with  a  broader  and  a  narrower  side,  may  be 
trit  d  on  each  of  its  sides  separately,  and  thus  it  may  be  seen 
whether,  in  a  given  weight,  the  extent  of  surface  of 'contact  makes 
any  difference  ;  the  body  may  be  loaded  with  different  weights, 
and  hence  may  be  learned  the  influence  of  pressure  upon  friction ; 
the  body  may  be  tried  as  soon  as  it  is  laid  on  the  table,  and  after 
remaining  on  it  for  a  longer  or  shorter  time,  in  order  to  learn 
whether  this  circumstance  alters  the  friction  ;  different  kinds  of 
bodies  may  be  tried,  and  the  influence  of  different  materials  as- 
certained ;  and  finally,  by  dragging  the  body  off  the  table  with 
different  degrees  of  velocity,  the  relation  of  friction  to  velocity 
may  be  investigated. 

349.  From  experiments  like  the  foregoing,  endlessly  varied, 
the  following  conclusions  were  established : 

(1.)  In  a  given  body,  extent  of  surface  makes  no  difference  in 
regard  to  friction  ;  a  brick  laid  on  its  edge  meets  with  the  same 
resistance  from  this  cause  as  when  laid  on  its  side. 

(2.)  Friction  is  proportioned  to  the  pressure.  If  the  pressure 
of  the  brick  is  doubled  or  trebled  by  laying  weights  upon  it,  the 
amount  of  friction  will  be  increased  in  the  same  ratio. 

(3.)  Friction  is  increased  by  bodies  remaining  for  some  time  in 
contact  with  each  other.  In  some  cases  it  does  not  reach  its 
maximum  under  four  or  five  days.  This  principle  therefore  af- 
fects slow  motions  much  more  than  such  as  are  rapid.  In  the 
mutual  contact  of  metals,  the  friction  attains  its  maximum  almost 
instantaneously.  But  when  metal  rubs  against  wood,  or  one 
piece  of  wood  against  another,  the  friction  is  always  increased  by 
resting.  Two  pieces  of  wood  acquire  the  utmost  friction  in  an 
hour  or  two ;  while  iron  running  on  oak  will  have  its  friction 
augmenting  for  five  or  six  days.  The  application  of  a  coat  of 
tallow  seems  to  protract  the  limit  of  friction.  This  limit  is  at- 
tained by  the  greased  surfaces  of  iron  and  copper  in  four  minutes ; 
while  pieces  of  wood,  treated  in  the  same  way,  will  have  their 
friction  gradually  augmented  during  nine  or  ten  days.* 

(4.)  The  friction  is  less  between  surfaces  of  different  kinds  of 
matter,  than  between  those  of  the  same  kind.  Copper  slides  on 
copper,  or  brass  on  brass,  with  greater  difficulty  than  copper  on 

*  Leslie,  El.  Nat.  Phil.  I,  225. 


244  NATURAL  PHILOSOPHY. 

brass  ;  and  it  is  a  general  rule  never  to  let  two  substances  of  the 
same  hardness  move  upon  each  other.  To  this  rule  cast  steel  is 
said  to  form  the  only  exception  ;  in  other  cases,  pivots  revolve 
with  less  resistance  on  either  harder  or  softer  substances,  than 
upon  those  of  the  same  material  with  themselves.*  When  be- 
tween the  surfaces  of  wood  newly  planed,  the  friction  would  be 
equal  to  one  half  the  pressure,  and  when  between  two  metallic 
surfaces  it  would  be  equal  to  one  fourth,  between  the  wood  and 
metal  it  would  amount  to  only  one  fifth  the  pressure. f 

(5.)  Friction  is  much  greater  at  the  first  moving  of  a  load,  than 
after  it  is  brought  freely  into  motion.  In  many  instances,  it  is 
reduced,  when  a  body  has  attained  its  final  velocity,  to  less  than 
one  half  of  what  it  was  at  first.J  With  regard  to  different  de- 
grees of  velocity  over  a  given  space,  it  is  a  general  principle,  that 
the  friction  is  the  same  for  all  velocities ;  that  a  carriage,  for  ex- 
ample, in  travelling  from  one  place  to  another,  would  encounter 
the  same  resistance  from  friction,  whether  it  performed  the  jour- 
ney in  one  hour  or  in  ten.  The  amount  of  friction,  however,  is 
augmented  in  very  slow  motions,  and  greatly  diminished  in  those 
that  are  very  swift.  In  this  instance,  the  increase  in  the  one 
case  and  the  diminution  in  the  other,  appears  to  have  some  rela- 
tion to  the  principle,  that  the  friction  of  bodies  is  increased  by 
their  remaining  in  contact.  From  some  observations  of  Pro- 
fessor Playfair,  made  at  the  slide  of  Alpnach,  where  large  fir- 
trees  are  carried  with  great  velocity  down  an  inclined  plane 
eight  miles  in  length,  it  would  appear  that,  in  the  case  of  very 
great  velocities,  friction  is  not,  according  to  the  common  doctrine 
either  proportioned  to  the  pressure,  or  independent  of  the  velo- 
city ;  but  that  the  ratio  to  the  pressure  is  greatly  diminished,  and 
the  actual  resistance  is  far  less  than  at  common  velocities. 
Thus,  none  but  large  trees  could  descend  the  plane  at  all ;  and 
when  a  tree  broke  into  two  pieces,  the  larger  part  would  pro- 
ceed while  the  smaller  would  stop  ;  and  the  trees  acquired  in 
their  descent  a  rapidity  of  motion,  incompatible  with  the  sup- 
position that  "  friction  acts  as  a  uniformly  retarding  force,"  which 
has  been  considered  as  an  established  principle. § 

The  foregoing  considerations  are  in  favor  of  rapid  travelling, 
whether  on  common  roads  or  on  railways,  since  the  amount  of 
the  resistance  is  so  much  less  than  in  slow  movements  ;  and  ac- 
cordingly it  is  said  that  the  great  speed  given  to  stage  coaches 
in  England,  amounting,  in  some  instances,  to  10  or  12  miles  per 
hour,  has  not  been  attended  with  the  degree  of  exhaustion  to  the 
teams  that  would  have  been  anticipated.)! 


*  Allen's  Mechanics,  p.  137.  t  Leslie.  t  Leslie,  Nat.  Phil.  I,  218. 

§  Playfair's  Works,  Vol.  I,  or  Ed.  Phil  Jour  VI,  345     Also  see  Nicholson's  Oper 
Mech.  II,  225. 
|]  Nich.  Oper.  Mech.       , 


MECHANICS.  245 

350.  The  angle  at  which  an  inclined  plane  must  be  elevated, 

in  order  that  a  given  body  resting  on  it  may  be  on  the  point  of 

sliding,  is  called  the  angle  of  friction,  or  sometimes  the  angle  of 

repose.     Let  ABC  (Fig.  167)  be  the  inclined  plane,  and  let  FD, 

C  Fig.  167. 


B  A 

parallel  to  BC,  represent  the  whole  weight  of  the  body.  FD 
being  resolved  into  FE  perpendicular  to  the  plane,  and  ED  coin- 
cident with  it,  ED  will  be  the  force  with  which  the  body  tends 
to  slide  down  the  plane,  and,  of  course,  that  which  is  resisted  by 
the  friction,  and  hence  it  may  be  taken  as  the  representative  of 
the  friction.  But  ED  is  to  FE  as  BC  to  BA,  or  as  the  height  of 
the  plane  to  the  base,  or  as  the  tangent  of  the  angle  of  friction 
to  radius.  Consequently,  putting  F  for  the  friction,  P  for  the 
whole  pressure  on  the  plane,  and  a  for  the  angle  of  friction,  then, 

F  :  P  : :  tan.  a  :  rad.  .*.  F=Pxtan.  a ; 

that  is,  the  friction  is  always  such  a  part  of  the  entire  pressure, 
as  is  denoted  by  the  product  of  that  pressure  into  the  tangent  of 
the  angle  of  friction.  This  angle  often  determines  the  figure 
which  natural  objects  spontaneously  assume.  Hence,  sand  hills 
are  more  sloping  than  eminences  composed  of  ordinary  mould, 
the  movable  particles  arranging  themselves  in  obedience  to  the 
foregoing  law.  The  angle  of  friction  of  iron  pressing  on  iron, 
is  found  to  be  16  degrees.  The  angle  given  to  the  threads  of  a 
screw,  is  regulated  on  this  principle.* 

351.  The  laws  of  friction  in  rolling  bodies  were  ascertained 
by  Coulomb,  by  comparing  the  forces  necessary  to  roll  a  cylinder 
upon  a  table  under  various  circumstances  ;  and  by  similar  ex- 
periments, were  found  the  modes  in  which  friction  takes  place 
in  bodies  revolving  on  an  axis.     The  comparative  loss  of  power 
in  these  three  cases  is  as  follows  : 

Friction  of  the  sliding  body  equal  to  £  the  pressure,  or  25  per  ct 
do.  revolving  do.  15 

do.  rolling       do.  5 

In  the  case  of  hollow  cylinders  revolving  on  an  axis,  the  lever- 
age of  the  wheel  aids  in  overcoming  friction,  as  has  been  already 
explained  in  Art.  306. 

352.  Friction  wheels,  a  contrivance  by  which  friction  is  dimin- 

*  Leslie. 


246  NATURAL    PHILOSOPHY. 

ished  in  the  greatest  degree  possible,  owe  their  efficacy  in  part 
to  the  operation  of  the  same  principle.*  Here  the  axis  of  a 
wheel,  instead  of  revolving  in  a  hollow  cylinder,  or  instead  of 
rubbing  against  a  fixed  surface,  rests,  at  each  of  its  extremities, 
on  the  circumference  of  two  wheels  placed  close  by  the  side  of 
each  other,  with  their  circumferences  intersecting.  The  axis 
rests  at  the  point  of  intersection,  and  as  it  revolves,  the  wheels 
revolve  with  it  with  the  same  velocity,  and  thus  all  friction  be- 
tween them  and  the  axis  is  prevented ;  and  what  remains  in  the 
machine  in  consequence  of  the  weight  of  the  wheels  themselves 
is  transferred  to  the  axles,  and  therefore  (Art.  300)  is  diminished 
in  the  ratio  of  the  diameter  of  one  of  the  wheels  to  that  of  its 
axis.  This  combination  may  be  repeated  by  several  pairs  of 
friction  wheels.  Eight  wheels  would  contract  the  friction  to  the 
thousandth  part.f 

353.  Other  more  common  methods  of  diminishing  friction  are, 
by  rendering  the  surfaces  smooth,  by  .using  rollers,  and  by  lubri- 
cating the  parts  in  contact.     The   amount  of  friction   in  the 
several  mechanical  powers  is  very  different.     In  the  lever  it  is 
very  small,  especially  when  the  turning  edge  is  of  hardened 
steel^  and  shaped  like  a  knife  or  prism,  and  turns  upon  a  hard 
and  smooth  basis.     The  wheel  and  axle  acting  upon  the  same 
principle  as  the  lever,  occasions  but  little  friction.     The  stiffness 
of  the  cordage,  however,  and  the  friction  of  the  gudgeons J  of 
the  axis,  have  an  effect  in  most  cases  equal  to  about  8  or  10  per 
cent,  of  the  entire  resistance.     The  pulley  is  attended  with  great 
loss  from  this  source.     It  is  rarely  less  than  20  per  cent,  and  often 
exceeds  60.     The  inclined  plane  involves  but  little  friction  when 
bodies  simply  roll  on  it ;  but  when  heavy  bodies  rest  on  axes,  as 
in  wheel  carriages,  the  resistance  from  friction  takes  place  in 
the  same  manner  as  upon  plane  surfaces.     The  transportation 
on  inclined  planes,  as  railways,  is  usually  by  means  of  wheels, 
since  the  resistance  to  sliding  movements  is  too  great  to  permit 
the  use  of  them.     The  screw  is  attended  with  a  great  amount 
of  friction.     Those  with  sharp  threads  have  more  than  those 
with  square  threads,  and  the  endless  screw  has  most  of  all.§    In 
both  the  screw  and  the  wedge,  the  friction  evidently  exceeds  the 
resistance  ;  otherwise  they  would  not  retain  their  position. 

354.  Friction  is  not,  therefore,  in  all  cases  to  be  considered  as 
unfavorable  to  the  operation  of  machinery.     It  is,  in  many  in- 
stances, a  highly  useful  force.     Many  structures,  as  those  of  brick 
and  stone,  owe  no  small  part  of  their  stability  to  the  roughness 
of  the  materials  of  which  they  are  composed ;  without  this  re- 

*  A  fine  example  of  the  application  of  friction  wheels  is  seen  in  Atwood's  Machine. 

t  Leslie,  El.  Nat.  Phil.  I,  233. 

t  Pivots  when  large  take  the  name  of  gudgeon?.  §  Emerson. 


MECHANICS.  247 

sistance,  the  screw  and  the  wedge  would  lose  their  efficacy,  and 
wheels  could  not  advance,  nor  could  animals  walk  on  the  ground; 
and  nails  would  lose  their  power  of  binding  separate  parts  to- 
gether. The  art  of  polishing  surfaces  depends  on  the  same 
cause,  and  the  edges  of  most  cutting  instruments  are  saws,  the 
teeth  of  which  are  more  or  less  fine,  and  act  on  a  similar  princi- 
ple. Even  in  certain  rotary  motions,  friction,  by  a  rope  or  other- 
wise, becomes  a  moving  force,  and  urges  a  body  in  particular  di- 
rections contrary  to  the  force  of  gravity. 

The  resistance  which  moving  bodies  sustain  from  the  air,  and 
from  water,  will  be  considered  hereafter. 


CHAPTER  VI. 

OF  PROJECTILES  AND  GUNNERY. 

355.  EARLY  in  the  17th  century,  Galileo,  a  celebrated  Italian 
philosopher,  established  the  mathematical  theory  of  projectiles, 
as  given  in  the  former  part  of  this  work.     Since  missiles  thrown 
by  instruments  of  warfare  are  among  the  number  of  projectiles, 
these  principles  were  supposed  to  constitute  the  foundation  of 
the  art  of  gunnery;  and  several  of  the  kings  of  Europe  held 
out  munificent  encouragement  to  experiments  in  this  department 
of  science.*     Galileo  had  indeed  intimated  that  the  resistance  of 
the  air,  (which  is  not  taken  into  the  account  in  the  mathematical 
theory  of  projectiles,)  might  occasion  some  deviation  in  bodies 
from  the  curve  of  a  parabola ;  but  this  fluid  appears  so  light  in 
comparison  with  the  heavy  metals  of  which  balls  are  made, 
being  10,000  times  lighter  than  lead,  that  for  a  long  time  after- 
ward it  was  totally  neglected. 

356.  About  the  year  1740,  Mr.  Robins,  an  eminent  English 
mathematician,  instituted  a  series  of  experiments  on  gunnery, 
which  showed  that  the  parabolic  theory  of  projectiles  is  so  modi- 
fied by  the  resistance  of  the  air,  as  to  be  wholly  inapplicable  to 
practice.     Experiments  on  the  same  subject  have  since  been  per- 
formed by  Count  Rumford,  and  by  Dr.  Hutton  of  the  Royal  Mili- 
tary Academy  at  Woolwich,  which  have  confirmed  the  results 
obtained  by  Mr.  Robins.     By  the  labors"  of  these  several  gentle- 
men, the  parabolic  theory  of  projectiles  has  received  its  proper 
modifications. 

*  Robison's  Mech.  Phil.  I,  167. 


248  NATURAL   PHILOSOPHY. 

357.  It  is  ascertained,  in  general,  that  projectiles  moving  slowly, 
describe  curves  which  are  nearly  parabolas  ;  while  such  as  move 
swiftly,  deviate   very  far  from  this  curve.     Indeed,  the  curve 
which  a  body  moving  with  great  velocity  in  the  air  describes,  is 
so  complicated,  that  the  utmost  resources  of  the  calculus  have 
hardly  been  able  to  investigate  its  nature  ;*  and  it  is  a  remarka- 
ble fact,  that  we  understand  the  motions  of  the  heavenly  bodies 
better  than  we  do  those  of  a  cannon  ball,  and  can  trace  the  path 
of  a  planet  better  than  we  can  that  of  an  arrow.     The  parabolic 
figure  described  in  the  case  of  projectiles  moving  slowly,  may 
be  observed  in  tracing  the  path  of  a  small  stone  thrown  into  the 
air,  and  more  especially  in  the  curves  described  by  jets  of  water, 
spouting  upward,  as  in  fountains.     But  when  the  jet  of  water  is 
more  rapid,  and  spouts  at  a  high  angle,  as  45  degrees  for  ex- 
ample, we  can  plainly  see  that  the  curve  deviates  greatly  from  a 
parabola.     The  remote  branch  of  the  curve  is  seen  to  be  much 
less  sloping  than  the  rising  branch  ;  and  even  in  very  great  jets, 
which  are  to  be  seen  in  some  great  water-works,  the  falling 
branch  is  almost  perpendicular  at  its  remote  extremity ;  and  the 
highest  point  of  the  curve  is  far  from  being  in  the  middle  be 
tween  the  spout  and  the  place  where  the  water  falls.     This  un- 
equal division  of  the  curve  by  its  highest  point  may  also  be  ob- 
served in  the  flight  of  an  arrow  or  a  bomb-shelLf 

358.  The  following  facts  also  show  the  discordance  between 
the  parabolic  theory  of  gunnery  and  experience.     A  cannon  ball, 
fired  in  such  a  direction  and  with  such  a  velocity,  that  its  random 
or  horizontal  range  ought  to  be  24  miles,  comes  to  the  ground 
short  of  one  mile.     The  times  of  rising  and  falling,  if  that  the- 
ory held  good,  ought  to  be  equal ;  but  the  time  of  rising  is  greater 
than  that  of  falling  at  great  elevations,  and  at  small  elevations, 
less  than  that  of  falling.     According  to  the  theory,  the  greatest 
random  is  at  an  angle  of  elevation  of  45  degrees,  (Art.  191,)  but 
in  practice  it  is  found  to  be  much  below  this.     The  greatest  ran- 
dom of  an  arrow  is  when  the  elevation  is  about  36  or  38  degrees. 
Indeed,  the  angle  for  the  greatest  horizontal  range,  may  be  at  all 
degrees  from  45°  to  30° ;  the  slowest  motions  and  the  largest 
shot  being  almost  at  45°,  but  gradually  more  and  more  below 
that  degree  as  the  shot  is  smaller  and  the  velocity  is  greater ;  till 
at  length,  with  the  most  rapid  motions  and  the  smallest  shot,  the 
angle  is  little  above  30  degrees.J     The  following  experiments 
were  made  in  France  by  Borda,  with  a  24  pounder,  with  the 
same'  charge  of  powder  in  each  experiment. 


»  Button's  Tracts,  No.  37.  t  Robison's  Mech.  Phil.  I,  185. 

t  Hut  ton,  Tract  37. 


MECHANICS.  249 

Elevation.  Range. 

15°  .....  1950 

30 2235 

45 2108 

60      .         '..''.''          .  .  .       1700 

75  .        , /r/><         ...  950 

Whence  it  appears  that  at  the  elevation  of  15  and  75,  the  ran- 
doms, instead  of  being  the  same,  (being  equally  distant  from  45.) 
were  as  the  numbers  1950  and  950.* 

359.  All  this  discordance  between  theory  and  practice  is  owing 
to  the  resistance  of  the  air,  which,  when  the  projectile  moves 
with  great  velocity,  becomes  enormous.     Nor  will  it  be  difficult, 
on  a  little  reflection,  to  comprehend  the  reason  why  this  resistance 
should  be  so  great.     The  force  with  which  a  projectile  strikes 
the  air  at  rest,  is  the  same  as  that  with  which  the  air  moving 
with  equal  velocity  would  strike  the  body  at  rest.     This,  in  the 
case  of  a  cannon  ball,  would  greatly  exceed  the  most  violent 
hurricane.     Again,  as  a  ball  moves  through  the  air,  it  displaces, 
that  is,  gives  motion  to,  great  quantities  of  air ;  yet  whatever 
motion  it  imparts  to  other  bodies  is  extinguished  in  itself.     The 
loss  of  motion  therefore  increases  very  fast  with  the  velocity.     It 
is  said  to  be  in  general  as  the  square  of  the  velocity :  so  that  a 
body  moving  through  the  air  w^n  ten  times  the  velocity  of  an- 
other body,  would  encounter  one  hundred  times  as  much  resist- 
ance.    In  very  swift  motions,  the  resistance  was  ascertained  by 
Robins  to  be  even  much  greater  than  in  the  ratio  of  the  square 
of  the  velocity. 

360.  The  researches  of  Mr.  Robins  were  made  chiefly  by  the 
aid  of  an  instrument  of  his  own  invention,  called  the  Ballistic 
Pendulum.     It  consists  of  little  more  than  a  large  block  of  wood, 
like  a  log,  suspended  after  the  manner  of  a  pendulum.     Now  if 
a  bullet  be  fired  into  the  block,  as  the  bullet  will  be  stopped,  and 
as  it  imparts  to  the  block  whatever  motion  it  loses,  consequently 
the  momentum  of  the  block,  after  the  stroke,  is  precisely  that  of 
the  ball  before  the  stroke.     Hence  the  weight  of  the  block  and 
that  of  the  ball  being  known,  and  the  velocity  imparted  to  the 
block  being  readily  determined  by  observation,  it  is  easy  to  find 
the  velocity  of  the  ball ;  for  the  weight  of  the  ball  is  to  the 
weight  of  the  block,  as  the  velocity  of  the  block  is  to  the  velo- 
city of  the  ball.f 

*  Robison's  Mech.  Phil.  I,  183. 

t  That  is,  putting  W,  w,  for  the  weight  of  the  block  and  ball  respectively,  and 
V,  t5,  for  their  velocities,  then  WxV=w>X»,  and  w  :  W  : :  V:».  In  other  experi- 
ments on  this  subject,  the  gun  itself  has  sometimes  been  suspended  like  a  pendulum, 
and  the  velocity  of  the  ball  estimated  by  the  distance  to  which  the  gun  recoiled,  since 
action  and  reaction  are  equal,  and  in  opposite  directions. 

32 


250  NATURAL    PHILOSOPHY. 

Example. — On  firing  an  8  pound  cannon  shot  into  a  ballistic 
pendulum  that  weighed  500  pounds,  the  pendulum  was  made  to 
move  over  a  space  of  16  feet  per  second :  What  was  the  velocity 
of  the  ball  ?  Ans.  1000  feet  per  second. 

361.  This  simple  apparatus  is   sufficient  for  ascertaining  a 
great  number  of  particulars  relative  to  the  art  of  gunnery.     If 
the  ball  is  fired  nearly  in  contact  with  the  block,  we  find  with 
what  velocity  it  leaves  the  gun  ;  if  at  different  distances  from 
the  block,  we  find  how  much  the  velocity  is  retarded  by  passing 
through  the  air,  for  those  distances  respectively.     If  at  a  given 
distance  we  vary  the  charge  of  powder,  we  find  the  respective 
changes  which  the  velocity  undergoes,  and  hence  learn  the  ratio 
that  ought  to  be  observed  between  the  powder  and  the  ball,  in 
order  to  produce  the  maximum  effect.     The  effects  resulting 
from  variations  in  the  length,  shape,  and  bore  of  the  gun,  are 
also  ascertained  with  equal  facility. 

362.  The  following  are  some  of  the  practical  results  ascer- 
tained by  the  experiments  of  Mr.  Robins,  Count  Rumford,  and  Dr. 
Hutton.     A  musket  ball,  discharged  with  a  common  charge  of 
powder,  issues  from  the  muzzle  of  the  piece  with  a  velocity  be- 
tween 1600  and  1700  feet  in  a  second.*    The  utmost  velocity  that 
can  be  given  to  a  cannon  ball  is  a  little  more  than  2000  feet  per 
second,  and  this  it  has  only  at  the  moment  of  leaving  the  gun. 
In  order  to  increase  the  velocity  from  1600  to  2000  it  requires 
half  as  much  more  powder,  which  involves  a  hazardous  strain 
upon  the  gun,  and  the  velocity  will  be  reduced  to  1300  before 
the  ball  has  proceeded  500  yards,  f 

363.  Mr.  Robins,  in  the  course  of  his  experiments  on  the  re- 
sistance of  the  air,  made  a  curious  observation.     In  very  moder- 
ate velocities,  the  retardations  were  nearly  as  their  squares.     As 
the  velocities  were  increased,  the  resistances  increased  at  a  some- 
what greater  rate,  but  with  a  certain  observable  regularity,  till 
the  velocity  exceeded  1100  feet  per  second.     But  when  the  velo- 
city is  increased  from  1100  to  1200,  the  increase  of  resistance  is 
prodigious.     After  this,  the  resistance  goes  on  increasing  nearly 
with  its  former  regularity.     Mr.  Robins  accounts  for  this  singu- 
lar fact  in  the  following  manner.     As  the  ball  rushes  through  the 
air,  the  air  falls  in  behind  it,  being  pressed  in  by  the  weight  of 
the  surrounding  air.     But  the  ball  may  move  so  rapidly  that  the 
air  cannot  instantaneously  fill  up  the  place  left  by  the  ball.     In 
this  case  the  ball  is  retarded,  not  only  by  the  resistance  of  the 

*  Space  fallen  through  to  acquire  the  velocity  of  1600  feet  per  second=i-— — 

(Art.  34,  p.  38,)  =40,000  nearly =1.6  miles 
t  Robison's  Mech.  Phil.  I,  201. 


MECHANICS.  251 

air  which  it  displaces,  but  also  by  its  having  the  pressure  of  the 
atmosphere  on  one  side  and  not  on  the  other.  It  is  found  by  cal- 
culation, that  the  velocity  with  which  air  rushes  into  a  vacuum, 
is  about  1100  or  1200*  feet  per  second  ;  consequently,  it  is  when 
the  ball  is  moving  at  this  rate,  that  it  loses  the  entire  pressure  of 
the  atmosphere  behind  it.f  It  is  remarkable  (says  Dr.  Robison) 
that  this  is  also  nearly  the  velocity  of  sound,  and  there  is  an  obser- 
vation of  this  kind,  which  seems  to  have  some  connection  with 
the  mechanical  fact  observed  by  Mr.  Robins.  If  a  person  stand 
in  such  a  direction  from  a  cannon,  when  it  is  discharged,  that  the 
ball  may  pass  him  at  no  great  distance,  he  will  hear  the  noise 
made  by  the  ball  rushing  through  the  air  at  the  time  of  its  flight, 
and  as  the  ball  approaches  him,  the  noise  should  become  more 
audible.  But  he  will  hear  the  noise  loudest  at  the  very  first,  im- 
mediately following  the  report  of  the  gun  ;  and  after  about  two 
seconds,  he  may  observe  the  sound  change  all  at  once,  and  not 
only  become  more  faint,  but  even  change  its  kind,  after  which 
the  sound  increases  as  the  ball  comes  nearer.  It  seems  highly 
probable  that  this  abrupt  alteration  in  the  sound  takes  place  just 
at  the  time  when  the  resistance  undergoes  such  a  change  ;  and 
that  it  is  owing  to  the  difference  in  the  nature  of  the  undulations, 
when  there  is  a  void  behind  the  ball,  and  when  there  is  not.J 

364.  From  the  foregoing  considerations  it  is  inferred  that  great 
charges  of  powder  are  absolutely  useless  in  the  service  of  artillery, 
especially  wrhen  the  distance  of  the  object  is  considerable,  and  that 
a  velocity  exceeding  1100  should  not  be  aimed  at.  The  maxi- 
mum service  charge  is  f  the  weight  of  the  ball.  In  close  naval 
engagements  great  velocities  are  injurious,  for  the  ball  may  then 
pass  through  both  sides  of  the  vessel  without  lodging,  and  the 
number  of  splinters  produced  by  a  ball  in  rapid  motion,  is  much 
less  than  is  caused  by  one  moving  more  slowly.  By  reducing 
the  charge  we  may  also  reduce  the  size  and  strength  of  the  gun  ; 
and  hence  guns  are  rrfade  of  smaller  dimensions  now  than  for- 
merly, in  order  to  do  the  same  execution.  The  velocity  with 
which  a  charge  of  powder  expands  itself  at  first,  is  estimated  by 
Hutton  as  high  as  5000  feet  per  second.§  As  it  expands,  this 
velocity  is  of  course  constantly  diminishing,  but  will  exceed  that 
of  the  ball  while  the  latter  is  passing  through  the  barrel  of  the 
gun,  and  will  act  as  a  constantly  accelerating  force.  Long  guns 
therefore  give  to  balls  a  greater  velocity  than  short  ones  ;  but  the 
gain  secured  in  this  way  after  a  moderate  length  is  so  small,|| 

*  The  velocity  with  which  air  begins  to  rush  into  a  void,  has  been  estimated  at 
1338  feet.  (See  Cambridge  Mechanics,  p.  377.) 

t  Robins,  Tract  on  Gunnery,  p.  181.  t  Robison's  Mech.  Phil.  I,  200. 

§  Hutton,  Tract  37. 

||  The  random,  according  to  Dr.  Hutton,  increases  only  as  the  fifth  root  of  the 
length,  which  is  so  small  an  increase  as  to  amount  to  only  about  a  seventh  part  more 
range  for  a  double  length  of  gun.  (Hutton,  Tract  37.) 


252  NATURAL   PHILOSOFEV. 

(there  being  also  some  disadvantages  peculiar  to  long  guns,)  that 
cannon  have  of  late  years  been  much  shortened.  In  the  naval 
service,  carronades  have  been  introduced.  These  are  a  short 
kind  of  gun,  with  small  bore,  requiring  for  a  charge  of  powder, 
only  one  twelfth  the  weight  of  the  ball.  Their  weight  and 
thickness  are  proportionally  reduced,  yet  in  close  action  they 
produce  effects  superior  to  those  of  long  guns.* 

365.  It  has  been  found  that  no  difference  is  caused  in  the  ve- 
locity or  range,  by  varying  the  weight  of  the  gun,  nor  by  the 
use  of  wads,  nor  by  different  degrees  of  ramming,  nor  by  firing 
the  charge  of  powder  in  different  parts  of  it ;  but  that  a  very 
great  difference  in  the  velocity  arises  from  a  small  degree  in 
the   windage.^      Indeed,  with  the    usual    established  windage 
only,  viz.  about  j\  of  the  calibre,  no  less  than  between  |  and  I 
of  the  powder  escapes  and  is  lost,  and  as  the  balls  are  often 
smaller  than  the  regulated  size,  it  frequently  happens  that  half 
the  powder  is  lost  by  unnecessary  windage.J      To  this  cause 
also,  namely,  too  great  windage,  Dr.  Hutton  ascribes  a  great  part 
of  the  sideways  deviation  of  a  ball ;  since,  if  in  passing  through 
the  barrel  of  the  gun,  it  is  knocked  from  side  to  side,  it  will  finally 
take  the  last  direction  which  it  happened  to  have  at  the  muzzle 
of  the  gun.§     Another  cause  of  this  deviation  from  the  line  of 
direction,  arises  from  a  want  of  perfect  sphericity  in  the  ball,  by 
which  means  the  two  sides  do  not  meet  with  equal  resistance. 
Rifles  owe  their  superiority  over  common  guns,  chiefly  to  their 
obviating  this  deviation.     They  have  a  spiral  groove  cut  in  their 
bore,  making  about  a  turn  and  a  half  in  the  whole  length  of  the 
barrel.     The  ball,  which  is  made  to  fit  close  to  avoid  too  great 
windage,  has  a  corresponding  motion  impressed  on  it,  which  it 
retains  after  it  leaves  the  gun,  continuing  to  revolve  around  the 
line  of  direction.     Whatever  inequalities,  therefore,  may  exist  in 
the  ball,  their  effects  are  neutralized,  by  their  being  first  on  one 
side  and  then  on  the  other  of  this  line. 

366.  When  a  ball  is  projected  from  a  piece  of  ordnance  at  a 
small  angle  of  elevation,  and  falls  upon  water,  or  on  a  plane  of 
hard  earth,  its  flight  will  not  cease,  but  it  will  rise  again  and  de- 
scribe a  second  curve  similar  to  the  first,  but  less ;  and  it  will 
continue  to  rebound,  until  the  whole  of  its  projectile  velocity  is 
destroyed.    This  species  of  firing  is  called  Ricochet.    It  is  applied 
with  great  advantage  from  sea-coast  batteries  upon  shipping,  and 
in  the  attack  of  fortresses.     The  pieces  are  fired  with  small 
charges  of  powder  and  elevated  only  from  3  to  6  degrees.     The 

*  Renwick,  Heads  of  Lectures,  in  Literary  and  Scientific  Repos.  IV,  281. 
t  By  windage  is  meant,  the  difference  between  the  diameter  of  the  ball  and  that  of 
the  bore  of  the  gun. 

J  Hutton,  Tract  3,  255.  §  Hutton,  Tract  38,  Prob.  4. 


MECHANICS.  253 

word  signifies  duck  and  drake,  or  rebounding ;  because  the  ball  or 
shot  thus  discharged,  goes  bounding  and  rolling  along,  killing  or 
destroying  every  thing  in  its  way,  like  the  bounding  of  a  flat  stone 
along  the  surface  of  water  when  thrown  almost  horizontally.* 


CHAPTER  VII. 

APPLICATIONS  OF  THE  PENDULUM. 

367.  THE  pendulum  has  three  most  important  and  interesting 
uses,  viz.  as  affording  a  measure  of  time,  the  means  of  determin- 
ing the  figure  of  the  earth,  and  a  standard  of  weights  and  meas- 


368.  Time  is  any  portion  of  indefinite  duration,  from  which  it 
may  be  considered  as  separated,  as  a  given,  number  of  yards  of 
cloth  are  measured  off  from  a  piece  of  unknown  length.     For 
the  measure  of  time  we  may  employ  any  instruments  or  marks 
that  divide  it  into  equal  portions.     The  period  occupied  by  a 
given  quantity  of  sand  in  running  through  a  funnel,  as  in  the 
hour-glass — the  successive  intervals  through  which  the  surface  of 
a  column  of  water  descends  while  discharging  itself  by  an  aper- 
ture in  the  bottom  or  side  of  the  vessel — the  beats  of  the  hand, 
as  in  music,  or  even  the  pulsations  of  the  wrist,  are  so  many  dif- 
ferent modes  of  measuring  time.     But  as  the  intervals  measured 
by  any  of  these  modes  are  not  perfectly  uniform,  and  as  they  are 
incapable  of  that  minute  subdivision  which  is  sometimes  required, 
none  of  them  is  adapted  to  the  purposes  of  the  astronomer,  who 
seeks  by  this  method  to  estimate  the  motions  of  the  heavenly 
bodies,  or  of  the  mariner,  who  depends  on  his  time-piece  for  the 
means  of  ascertaining  his  longitude  at  sea. 

369.  The  adaptation  of  the  pendulum  to  the  measuring  of  time, 
was  first  noticed  by  Galileo  on  observing  the  vibrations  of  a  lamp 
suspended  from  the  ceiling  of  a  church.     The  theoretical  doc- 
trines of  the  pendulum  revealed  the  fact,  that  all  the  vibrations 
of  a  pendulum  of  given  length,  are  performed  in  nearly  equal 
times,  and,  when  made  to  move  in  the  arc  of  a  cycloid,  (Art. 
180,)  in  times  that  are  exactly  equal.     Hence  the   pendulum 
becomes  a  measure  of  time.     Galileo,  indeed,  used  this  instru- 
ment by  itself,  counting  the  number  of  its  vibrations ;  but  Huy- 
gens,  an  astronomer  of  Holland,  first  connected  it  with  clock- 

*  Button's  Math,  and  Phi  Diet,  II,  374. 


254  NATURAL   PHILOSOPHY 

work,  by  which  its  motions  are  indefinitely  continued,  more  pre- 
cisely regulated,  and  the  number  of  its  vibrations  exactly  regis- 
tered. Pendulums  are  usually  made  of  such  a  length  as  to  beat 
seconds,  or  to  indicate,  at  each  vibration,  the  g^|o^  Par^  °f  a 
mean  solar  day  ;  but  as  the  period  occupied  by  the  passage  of  a 
star  from  the  meridian  round  to  the  same  meridian  again  (called 
a  sidereal  day)  is  less  than  that  occupied  by  the  sun,  so  the  pen- 
dulum regulated  to  solar  time,  may,  by  being  shortened  a  little, 
be  made  to  indicate  the  corresponding  aliquot  parts  of  a  sidereal 
day.  Since,  moreover,  the  times  of  vibration  are  as  the  square 
roots  of  the  lengths,  (Art.  181,)  a  pendulum  may  be  made  to  beat 
half-seconds  by  being  made  only  one  fourth  the  length  of  the 
seconds  pendulum,  or  to  beat  once  in  two  seconds  by  making  it 
four  times  as  long. 

370.  As  the  vibrations  of  the  pendulum  in  circular  arcs  are  not 
exactly  equal  to  each  other,  it  has  been  attempted  to  make  it  vi- 
brate in  the  arc  of  a  cycloid,  (Art.  179  ;)  but  the  practical  difficul- 
ties involved  in  making  the  pendulum  rod  adapt  itself  to  the 
cheeks  of  a  cycloid  are  so  great,  that  this  method  has  been  aban- 
doned in  practice,  and  is  at  present  regarded  only  as  a  theoretical 
curiosity.     Where  the  arcs  of  vibration  are  very  small,  they  will 
not  differ  sensibly  from  portions  of  a  cycloid,  and  the  times  of 
vibration  may  without  sensible  error  be  considered  as  equal.     In 
cases  where  extreme  accuracy  is  required,  the  exact  allowance, 
mathematically  determined,  may  be  applied  to  vibrations  per- 
formed in  circular  arcs  to  reduce  them  to  the  corresponding 
vibrations  in  a  cycloid  ;  and  a  still  further  allowance  may  be 
made  for  the  resistance  of  the  air,  so  as  to  make  the  vibrations 
correspond  to  such  as  they  would  be  if  performed  in  a  vacuum. 

371.  The  cycloid  is  the  curve  of  swiftest  descent ;  that  is,  a 
body  will  descend  from  a  given  height  in  less  time  on  this  than 
on  any  other  curve,  or  even  than  on  an  inclined  plane.     This 
proposition,  in  general,  is  demonstrated  by  the  aid  of  the  calculus ; 
but  the  cycloid  may  be  readily  compared  with  the  inclined  plane 
by  experiment,  as  in  the  annexed  figure,  where  two  balls  let  off 


Fig.  168. 


MECHANICS.  255 

at  the  same  instant  by  raising  a  joint  at  the  top  of  the  figure, 
roll  down  the  plane  and  the  curve  respectively.  The  ball  reaches 
the  bottom  soonest  by  way  of  the  curve.  The  velocity  acquired 
at  first  by  a  direction  more  nearly  perpendicular,  more  thau 
compensates  for  the  greater  length  of  the  course. 

Moreover,  if  both  the  balls  be  let  off  at  any  different  heights 
on  the  cycloid,  they  will  reach  the  lowest  points  at  the  same 
instant,  since  it  is  a  property  of  this  curve,  that  the  accelerating 
force  at  any  point  in  it  is  proportional  to  the  distance  of  that 
point  from  the  lowest  point ;  consequently,  the  upper  ball  being 
urged  by  a  force  as  much  greater  as  its  distance  is  greater, 
reaches  the  bottom  at  the  same  instant  with  the  lower  ball. 
On  the  same  principle  the  pendulum  performs  all  its  vibrations 
in  a  cycloid  in  equal  times. 

372.  The  pendulum  is  liable  to  some  sources  of  inaccuracy, 
which  it  has  cost  much  labor  and  skill  to  obviate.  The  rod  and 
bulb  are  both  subject  to  expand  by  heat  and  contract  by  cold. 
In  the  former  case  the  center  of  oscillation  (Art.  170)  is  carried 
too  far  from  the  center  of  motion,  that  is,  the  pendulum  becomes 
too  long,  and  the  clock  goes  too  slow ;  in  the  latter  case,  the 
pendulum  being  shortened,  the  clock  goes  too  fast.  With  a  pen- 
dulum having  an  iron  rod.  a  difference  of  temperature  of  25  de- 
grees would  make  a  difference  of  six  seconds  in  24  hours  in  the 
rate  of  the  clock.* 

Pendulums  so  constructed  as  to  remedy  these  defects  arising 
from  change  of  temperature,  are  called  Compensation  Pendu- 
lums. The  general  principle  of  these  instruments  is  as  follows  : 
we  connect  with  the  pendulum  rod,  some  substances  which  are 
made  to  expand  or  contract  in  a  direction  opposite  to  that  in 
which  the  rod  itself  contracts  or  expands,  and  thus  maintains  the 
center  of  oscillation,  at  a  uniform  distance  from  the  center  of  sus- 
pension. The  length  of  the  theoretical  or  simple  pendulum,  it 
may  be  remarked,  depends  wholly  on  the  distance  between  these 
two  points,  and  not  upon  the  length  of  the  pendulum  rod.  If 
then  we  make  the  pendulum  rod  of  one  kind  of  metal,  as  steel, 
and  connect  with  the  bulb  another  kind  of  metal,  as  brass,  which 
is  expanded  more  by  the  same  amount  of  heat,  a  less  length  of  the 
latter  expanding  upward,  will  compensate  for  a  greater  length  of 
the  former  expanding  downward,  and  the  center  of  oscillation  will 
be  kept  at  the  same  constant  distance  from  the  center  of  suspen- 
sion. Now  it  is  found  by  experiment  that  the  lengths  of  these 
substances,  which  are  equivalent  to  each  other,  are  as  1  :  .6091, 
or  as  100  :  61  nearly.  Therefore,  if  we  connect  61  inches  of  brass 
with  100  inches  of  steel,  the  former  so  arranged  as  to  expand 
upward  while  the  latter  expands  downward,  we  shall  have 

*  Kater. 


Fig.  169. 


256  NATURAL   PHILOSOPHY. 

a  compensation  pendulum.  The  first  instru- 
ment constructed  on  this  principle  is  called 
the  Gridiron  Pendulum,  a  name  which  it  de- 
rives from  the  fancied  resemblance  which  the 
parallel  bars  of  steel  and  brass  have  to  the 
gridiron.  In  the  following  figure,  the  five  bars 
marked  s  are  of  steel,  and  the  four  marked  b 
are  of  brass.  It  will  be  seen  by  the  figure  that 
the  rods  of  steel  can  elongate  themselves 
only  downward,  while  those  of  brass  can 
expand  only  upward ;  and  being  combined 
in  the  ratio  of  100  to  61,  they  will  exactly 
compensate  each  other.  A  clock  furnished 
with  the  gridiron  pendulum  is  found  capable 
of  keeping  very  accurate  time.  Indeed  a 
clock  of  this  kind,  constructed  by  Harrison, 
the  inventor  of  the  gridiron  pendulum,  gained 
the  great  premium  offered  by  the  British 
Board  of  Longitude,  for  a  time-keeper  which 
would  keep  time  for  a  given  period  to  a  cer- 
tain degree  of  accuracy. 

Another  form  of  the  compensation  pendulum 
consists  of  a  jar  of  mercury,  suspended  from 
the  rod  m  the  place  of  the  bulb.  By  means  of 
a  kind  of  stirrup  the  jar  is  so  connected  with 
the  rod  as  to  expand  upward  while  the  rod  expands  downward. 
The  mercury  being  a  very  expansible  metal,  the  length  of  the 
mercurial  column  required  for  the  compensation  of  a  steel  rod, 
of  42  inches,  (including  the  stirrup,)  is  only  6.31  inches.*  This 
compact  and  simple  form  of  the  compensation  pendulum,  makes 
this  variety  peculiarly  eligible. 

373.  In  stationary  time-pieces,  the  pendulum  affords  the  most 
eligible,  as  it  is  the  most  accurate,  measure  of  time.     But  where 
the  instrument  is  required  to  be  portable,  as  in  watches  and 
chronometers,  a   spiral   spring   called  the   main-spring   is   the 
moving  power,  instead  of  a  weight,  and  a  balance-wheel  takes 
the  place  of  the  pendulum.     The  hair-spring  attached  to  this 
maintains  its  vibrations,  coiling  up  in  one  direction  and  uncoiling 
in  the  opposite,  while  the  main-spring  supplies  the  power  in  the 
same  manner  as  the  weight  of  a  clock. 

374.  The  use  of  the  pendulum  in  investigating  the  figure  of  the 
earth,  results  from  its  power  of  measuring  the  intensity  of  the 
force   of  gravity  in  any  given  place.      (Art.    182.)      Now   as 
this  force  varies  inversely  as  the  square  of  the  distance  from  the 


Kater  in  Lardner's  Mechanics,  p.  331.     Daily,  Astron.  Trans.,  1824,  p.  381. 


MECHANICS.  257 

center  of  the  earth,  when  the  force  of  gravity  is  ascertained  at 
a  great  number  of  places  remote  from  each  other,  a  comparison 
of  these  observations  indicates  the  respective  distances  of  these 
places  from  the  center  of  the  earth,  and  of  course,  when  the  ob- 
servations are  sufficiently  multiplied,  they  indicate,  collectively, 
the  figure  of  the  earth. 

If,  therefore,  one  were  to  start  from  the  equator  with  a  pendu- 
lum which  there  vibrated  seconds,  and  should  proceed  with  it  to 
the  north  pole,  and  there  count  the  number  of  vibrations  it  would 
make  in  an  hour,  he  might  thus  ascertain  the  respective  forces 
of  gravity  at  those  two  points,  and  hence  learn  the  ratio  between 
the  equatorial  and  polar  diameters.  For,  according  to  Art.  183, 
the  number  of  vibrations  performed  by  a  pendulum  in  any  given 
time,  are  as  the  square  roots  of  the  forces  of  gravity.  Although 
it  might  be  convenient,  were  it  practicable,  to  derive  the  ratio 
between  the  equatorial  and  polar  diameters  of  the  earth  directly 
from  observations  made  at  these  two  points,  yet  as  that  cannot 
be  done,  we  can  only  obtain  the  law  of  curvature,  and  hence 
derive  the  figure  of  the  earth,  by  observations  made  in  many  dif- 
ferent latitudes.  Such  observations  have  been  made  with  the 
greatest  accuracy  by  Kater,  Sabine,  and  others  ;  and,  although 
the  result  shows  that  the  figure  of  the  earth  is  that  of  an  oblate 
spheroid,  yet  the  difference  between  the  two  diameters,  or  the 
elUpticity  as  it  is  technically  called,  is  somewhat  greater  when 
determined  by  the  pendulum,  than  when  estimated  by  measuring 
the  length  of  the  respective  degrees  of  the  meridian  from  the 
equator  to  the  pole.  By  a  comparison  of  a  great  number  of 
pendulum  experiments,  Baily,  an  English  astronomer,  makes  the 
ellipticity  of  the  earth  ?Vs  °f  the  equatorial  diameter,  while 
that  derived  from  a  similar  comparison  of  measures  of  the  me- 
ridian, is  j^-g.  Hence,  the  pendulum  would  lead  us  to  the  con- 
clusion that  the  diameter  of  the  earth  when  taken  through  the 
equator  is  about  34  miles  greater  than  when  taken  from  pole  to 
pole,  while  actual  measurements  would  make  the  same  excess 
about  26  miles.* 

Measurements  of  arcs  of  the  meridian,  made  for  the  express 
purpose  of  determining  the  true  figure  of  the  earth,  have  been 
executed  with  an  astonishing  degree  of  accuracy,  in  various 
countries,  and  in  different  latitudes,  from  the  equatorial  regions 
to  points  within  the  Arctic  circle.  By  these  means,  the  nature 
of  the  curve  that  encompasses  the  earth  would  be  ascertained, 
and  the  figure  of  the  earth  determined.  The  combined  result 
of  all  these  measurements  is  as  already  stated. 

375.  A  third  important  application  of  the  pendulum  is,  as  a 
standard  of  linear  measures. 

*  Herschel's  Astronomy. 


258  NATURAL   PHILOSOPHY. 

In  order  to  insure  confidence  in  business  transactions,  it  is  very 
essential  that -the  weights  and  measures  employed,  should  be 
and  remain  of  a  certain  known  amount  or  length, — a  condition 
which  cannot  be  attained  otherwise  than  by  adapting  them  to  a 
fixed  and  invariable  standard.  To  fix  on  such  a  standard  is  a 
matter  of  more  difficulty  than  would  at  first  be  imagined.  All 
things  on  the  face  of  the  earth  are  more  or  less  subject  to  change, 
not  only  the  works  of  man,  but  even  those  of  nature.  The  ele- 
ments decompose  or  wear  down,  in  time,  the  hardest  rocks.  In 
ancient  times,  the  standards  of  weights  and  measures  were  de- 
rived from  the  parts  of  the  human  body ;  as,  a  hand's  breadth, 
a  cubit,  a  digit,  &c.  The  English  linear  measures  were,  for  a 
long  time,  referred  to  the  length  of  a  yard-stick  taken  from  the 
length  of  the  arm  of  Henry  the  Vllth,  and  preserved  in  the 
tower  of  London.*  Of  the  effectiveness  of  such  a  standard,  it 
is  sufficient  to  mention  the  impossibility  of  verifying  it  after  the 
death  of  the  king.  The  idea  of  standards  involves  two  condi- 
tions that  are  indispensable, — the  constancy  of  the  thing  itself, 
and  the  power  of  verification,  should  any  change  in  it  even  be 
suspected.  These  conditions  are  secured  by  connecting  the 
standard  with  the  immutable  laws  of  nature.  In  the  year  1790, 
the  French  government  undertook  to  effect  a  complete  change 
in  the  weights  and  measures  in  use  throughout  the  world,  and 
to  derive  an  unalterable  standard  from  the  dimensions  of  the 
earth  itself.  For  this  purpose,  they  undertook  the  determination 
of  the  exact  length  of  a  quadrant  of  the  meridian,  extending  from 
the  equator  to  the  North  Pole.  A  certain  aliquot  part  (the  ten 
millionth)  of  this  they  denominated  a  metre,  which  was  to  be 
the  unit  of  all  linear  measures.  The  square  of  this  furnished 
a  measure  of  surfaces  and  the  cube  a  measure  of  solids. 
Measures  of  weight  were  derived  from  the  weight  of  a  certain 
volume  of  water. 

376.  We  may  exemplify  the  precautions  necessary  in  order  to 
make  the  pendulum  an  accurate  standard  of  measures,  by  review- 
ing the  conditions  under  which  it  is  adopted  as  a  standard  in  the 
state  of  New  York. 

The  standard  is  the  pendulum  vibrating  seconds,  in  a  cycloidal 
arc,  and  in  a  vacuum,  in  Columbia  College  in  the  city  of  New 
York.  The  vibrations  are  required  to  be  in  a  cycloidal  arc,  be- 
cause those  performed  in  circular  arcs  are  not  absolutely  uniform. 
But  the  vibrations  performed  in  circular  arcs  may  be  reduced  to 
the  corresponding  cycloidal  arcs  upon  mathematical  principles. 
As  the  resistance  of  the  air  might  be  unequal  at  different  times, 
such  an  allowance  must  always  be  made  for  this,  as  will  neutral- 
ize its  effect.  A  particular  spot  is  designated,  because  local 

*  Adams  on  Weights  and  Measures. 


MECHANICS.  Sii> 

causes  have  some  influence  upon  the  vibrations  of  the  pendu- 
lum. 

The  unit  of  linear  measures  is  the  yard,  which  is  of  such  mag- 
nitude as  to  bear  to  the  pendulum  the  proportion  of  1  to  1.086158. 
Some  have  proposed  to  make  the  seconds  pendulum  itself  the 
unit  of  linear  measures,  and  to  make  all  other  measures  of  length 
parts  and  multiples  of  this  ;  but  it  is  thought  advisable  to  retain 
the  common  measures  to  which  the  habits  of  society  are  adapted, 
determining  their  exact  amount  by  referring  them  to  this  in- 
variable standard.  The  usual  subdivisions  of  the  yard  into  feet, 
inches,  and  so  on,  remain  as  they  are.  The  standard  temperature 
of  the  standard  yard-stick,  is  that  of  melting  ice.  It  is  necessary 
to  attend  to  this  circumstance,  because  all  bodies  are  expanded 
by  heat  and  contracted  by  cold,  so  that  the  yard-stick  is  of  a 
uniform  length  only  at  a  given  temperature. 

The  unit  of  measures  of  weight  is  the  avoirdupois  pound,  of 
such  magnitude  that  a  cubic  foot  of  pure  water,  at  its  maximum 
density,*  shall  weigh  1000  ounces,  or  62£  pounds. 

The  unit  of  dry  measures  of  capacity  is  the  gallon,  a  vessel  of 
such  magnitude  as  to  hold  exactly  10  Ibs.  of  pure  water,  at  its 
maximum  density.  The  bushel  therefore  holds  80  Ibs.  The 
unit  of  liquid  measures  is  also  a  gallon,  containing  eight  pounds 
of  distilled  water,  at  its  maximum  density.f 

The  government  of  the  United  States  has  adopted,  for  the 
different  standards,  the  following  bases. 

Standard  -of  Length.  The  yard  of  3  feet  or  36  inches,  from 
the  scale  of  Troughton,  which  is  a  brass  scale  of  82  inches,  made 
by  a  celebrated  English  artist  for  the  survey  of  our  coast.  Hence 
the  yard  adopted  as  the  standard,  is  identical  with  the  British 
imperial  standard. 

Standard  of  Weight.  The  troy  pound,  containing  5762.38 
grains,  used  as  a  standard  at  the  mint  of  the  United  States. 

Standard  of  Dry  Measure.  The  British  Winchester  bushel 
of  2150.4  cubic  inches,  equal  to  77.6274  pounds  of  distilled  wa- 
ter, at  the  maximum  density. 

Standard  of  Liquid  Measure.  The  English  wine  gallon  of 
231  cubic  inches,  equal  to  8.339  pounds  advoirdupois  of  distilled 
water,  at  the  maximum  density.J 

*  Water  expands  as  its  temperature  rises  above  or  falls  below  a  certain  point ; 
about  40°  of  Fahrenheit.  Hence  the  necessity  of  employing  it  as  a  standard  at  ita 
maximum  density. 

t  See  an  able  view  of  the  applications  of  the  Pendulum  in  Renwick's  Mechanics 

t  North  Amer.  Rev.  for  1837,  p.  290. 


NATURAL   PHILOSOPHY. 


PART   III. HYDROSTATICS. 


377.  THE  principles  of  Mechanics  demonstrated  and  explain- 
ed in  the  foregoing  pages,  are  universal  in  their  application,  ex- 
tending alike  to  all  bodies,  whether  solid  or  fluid.     But  in  addi- 
tion to  those  properties  which  fluids  have  in  common  with  solids, 
and  which  bring  them  under  the  general  laws  of  Mechanics,  they 
have  also  properties  peculiar  to  themselves,  and  which  give  rise 
to  a  distinct  class  of  mechanical  principles,  not  applicable  to 
solid  bodies.     These  are  embraced  under  the  heads  of  HYDRO- 
STATICS and  PNEUMATICS,  the  former  division  comprising  the  doc- 
trine of  liquids,  and  the  latter  that  of  aeriform  bodies  or  gases.* 

In  Mechanics,  after  having  ascertained  a  few  fundamental 
principles  by  experiment  and  observation,  the  superstructure  is 
raised  chiefly  by  mathematical  reasoning,  and  thus  the  great 
body  of  truths  in  that  science  are  established  ;  but  in  Hydrostatics, 
and  the  other  subjects  of  Natural  Philosophy  which  follow,  we 
are  much  more  dependent  on  experiment,  which  frequent^  affords 
us  more  satisfactory  evidence,  than  we  can  obtain  by  the  appli- 
cation of  abstract  mechanical  principles.  ^ 

378.  A  FLUID  is  a  body  whose  particles  move  easily  among  them- 
selves, and  yield  to  the  least  force  impressed;  and  which,  when  that 
force  is  removed,  recovers  its  previous  state.} 

In  accordance  with  the  example  of  Sir  Isaac  Newton,  which 
has  been  followed  by  most  writers  on  Hydrostatics,  we  have  in 
troduced  the  foregoing  definition  of  a  fluid.  Although  the  best 
perhaps  that  the  subject  admits  of,  yet  it  is  not  very  discrimina- 
ting, since  such  substances  as  quicksand,  or  the  powder  of  mag- 
nesia, have  their  particles  as  movable  as  those  of  tar  or  syrup, 
while  yet  the  former  are  solid  and  the  latter  fluid  bodies.  The 
fact  is,  that  no  definition  can  be  given  of  a  fluid  which  would 

*  In  some  treatises  these  subjects  are  distributed  under  the  heads  of  Hydrostatics 
and  Hydrodynamics,  the  former  comprehending  the  mechanical  properties  of  fluids 
at  rest,  and  the  latter  those  of  fluids  in  motion.  In  other  works,  what  relates  to 
liquids  or  non-elastic  fluids,  is  divided  into  Hydrostatics  and  Hydraulics,  (the  latter 
denoting  the  mechanical  powers  and  agencies  of  running  water  and  of  machines 
carried  by  water,)  and  Pneumatics,  The  most  scientific  division  is  that  adopted  in 
the  Edinburgh  Encyclopedia,  where  the  term  Hydrodynamics  (from  Y<5uif>  and 
A»i>a/<ij)  is  used  to  denote  in  general  the  mechanical  powers  and  agencies  of  fluids ; 
and  this  head  is  divided  into  the  two,  Hydrostatics  and  Hydraul:'cs.  Pneumatics  is 
treated  of  in  a  separate  article.  But  we  prefer  to  follow  the  example  of  those  who 
arrange  these  subjects  under  the  two  heads  specified  in  the  text. 

t  Vinoe. 


HYDROSTATICS.  261 

convey  to  one  unacquainted  with  these  bodies  any  adequate  idea 
of  the  distinction  between  them ;  and  it  may  be  doubtful  whether 
any  better  definition  can  be  given  of  a  fluid,  than  that  it  is  a 
generic  term,  comprehending  liquids  and  gases,  while  liquids  are 
defined  to  be  bodies  in  the  form  of  water,  and  gases  bodies  in  the 
form  of  air.  The  property  included  in  thr  definition  of  Profes- 
sor Vince,  wrhich  we  have  copied,  namely,  that  a  fluid  when  dis- 
turbed by  any  force  impressed  recovers  itself,  is  as  characteristic 
of  this  class  of  bodies  as  the  mobility  of  their  parts. 

Since  water,  wind,  and  steam,  are  the  only  fluids  that  are 
usually  employed  as  mechanical  agents,  the  doctrines  of  Hydro- 
statics and  Pneumatics  have  regard  chiefly  to  them ;  but  the 
principles  established  respecting  these,  are  applicable  also  to  all 
analogous  bodies. 

379.  It  has  been  usual  to  denominate  liquids  and  gases  re- 
spectively non-elastic  and  elastic  fluids,  on  the  supposition  that 
water  and  other  liquids  are  nearly  or  quite  incompressible.  An 
experiment  performed  by  the  Florentine  academicians,  as  long 
ago  as  1650,  seemed  to  prove  that  water  is  wholly  incompressible. 
They  filled  a  hollow  ball  of  gold  with  water,  and  subjected  it  to 
a  strong  pressure.*  The  water,  not  yielding  to  the  compression, 
oozed  through  the  pores  of  the  gold.  Considering  the  great  den- 
sity and  compactness  of  this  metal,  the  experiment  was  for  a  long 
time  held  as  proving  decisively  that  water  is  wholly  incompressi- 
ble. Although  this  experiment  shows  that  water  is  compressed 
with  great  difficulty,  yet  later  experiments  have  proved,  that  it 
is  still  capable  of  compression.  The  most  decisive  evidence  of 
this  point  has  been  recently  afforded  by  the  experiments  of  Mr. 
Perkins.  It  had  been  previously  ascertained  that  by  a  pressure 
equivalent  to  that  of  the  atmosphere,  or  about  fifteen  pounds  to 
the  square  inch,  water  is  compressed  about  one  part  in  twenty- 
two  thousand.  Mr.  Perkins,  by  methods  to  be  described  here- 
after, applied  successive  degrees  of  pressure  up  to  that  of  two 
thousand  atmospheres,  or  30,000  pounds  to  the  square  inch,  and 
found  the  contraction  of  volume  to  be  nearly  one  twelfth  of  the 
whole. 

With  these  preliminary  remarks,  we  may  now  enter  upon  the 
immediate  consideration  of  the  principal  subject  before  us. 

*  The  method  of  applying  the  pressure  is  said  to  have  been  by  means  of  a  screw 
working  through  a  water-tight  joint.— Partington.  As  the  screw  was  forced  into 
the  water,  the  latter  must  either  be  compressed  or  make  its  escape  from  the  ball. 
(Art.  2.)  An  easier  mode  of  applying  the  pressure  would  have  been,  to  put  the  ball 
into  a  vise  and  flatten  it,  by  which  its  capacity  would  have  been  diminished. 


NATURAL   PHILOSOPHY. 


CHAPTER  I. 

OF   LIQUIDS    OR  NON-ELASTIC   FLUIDS   AT  REST  OR   UN    EQUI- 
LIBRIUM. 

380.  HYDROSTATICS  is  that  branch  of  Natural  Philosophy  which 
treats  of  the  mechanical  properties  and  agencies  of  LIQUIDS. 

381.  Fluids  at  rest  press  equally  in  all  directions. 

A  point  in  a  mass  of  fluid,  taken  at  any  depth,  exerts  and  sus- 
tains the  same  pressure  in  all  directions,  upward,  downward, 
or  laterally.  This  is  the  most  remarkable  property  of  fluids,  and 
is  what  particularly  distinguishes  them  from  solids,  which  press 
only  downward,  or  in  the  direction  of  gravity.  This  property 
naturally  results  from  the  freedom  of  motion  that  subsists  between 
the  particles  of  fluids  ;  for  if,  when  a  fluid  is  at  rest,  the  pressure 
on  any"  given  portion  were  not  equal  in  all  directions,  that  portion 
would  move  in  the  direction  in  which  the  resistance  was  least. 
But  by  the  supposition  it  does  not  move  :  therefore  it  is  kept  at 
rest  by  equal  and  contrary  forces  acting  on  all  sides.  But  the 
most  satisfactory  evidence  of  this  truth  is  obtained  from  experi- 
ments. On  opening  an  orifice  in  the  side  of  a  vessel  of  water, 
and  estimating  the  force  with  which  the  water  issues,  it  is  found 
to  be  equal  to  the  weight  of  the  incumbent  fluid ;  and  the  upward 
pressure  of  water  at  a  certain  depth  is  found  to  sustain  the 
heaviest  bodies  when  exposed  to  its  action  alone,  the  column 
above  the  bodies,  and  of  course  the  downward  pressure,  being 
removed. 

382.  A  given  pressure  or  blow  impressed  on  any  portion  of  a 
mass  of  water  confined  in  a  vessel,  is  distributed  equally  through 
all  parts  of  the  mass. 

A  given  pressure,  as  that  made  by  a  plug  forced  inward  upon 
a  square  inch  of  the  surface  of  a  fluid  confined  in  a  vessel,  is  sud- 
denly communicated  to  every  square  inch  of  the  vessel's  surface, 
however  large,  and  to  every  inch  of  the  surface  of  any  body  im- 
mersed in  it.  Thus  if  I  attempt  to  force  a  cork  into  a  vessel  full 
of  water,  the  pressure  will  be  felt  not  merely  by  the  portion  of 
the  water  directly  in  the  range  of  the  cork,  but  by  all  parts  of 
the  mass  alike  ;  and  the  liabilit  y  of  the  body  to  break,  supposing 
it  to  be  of  uniform  strength  thi  oughout,  will  be  as  great  in  one 
place  as  in  another,  and  it  will  break  at  the  point  where  it  hap- 
pens to  be  the  weakest,  however  that  point  may  be  situated  rela- 
tively to  the  place  where  the  cork  is  applied ;  and  the  effect  will 


HYDROSTATICS.  263 

be  the  same  whether  the  stopper  be  inserted  at  the  top,  the  bot- 
tom, or  the  side  of  the  vessel. 

383.  It  is  this  principle  which  operates  FiS-  170. 

with  such  astonishing  effect  in  the  Hy- 
drostatic Press,  by  means  of  which  a 
single  man  can  exert  a  force  which  is 
adequate  to  crush  the  hardest  substances, 
or  to  cut  in  two  the  largest  bars  of  iron.* 
Its  construction  is  as  folio  ws.f  Fig.  170, 
represents  a  press  made  of  the  strongest 
timbers,  the  foundation  of  which  is  com- 
monly laid  in  solid  masonry.  AB  is  a 
small  cylinder  in  which  moves  the  piston 
of  a  forcing  pump,  and  CD  is  a  large  cylinder  in  which  also 
moves  a  piston,  having  the  upper  end  of  its  rod  pressing  against 
a  movable  plank  E,  between  which  and  the  large  beam  above  is 
placed  the  substance  to  be  subjected  to  pressure,  as  for  example  a 
pile  of  new-bound  books.  By  the  action  of  the  pump  handle,  water 
is  raised  into  the  small  cylinder,  and,  on  depressing  the  piston, 
it  is  forced  through  a  valve  at  B  into  the  larger  cylinder  .and 
raises  the  piston  D,  which  expends  its  whole  force  on  the  bodies 
confined  at  E.  .  Now,  since  whatever  force  is  applied  to  any  one 
portion  of  the  fluid,  extends  alike  to  every  part,  therefore  the 
force  which  is  exerted  by  the  pump  upon  the  smaller  column,  is 
transmitted  unimpaired  to  every  inch  of  the  larger  column,  and 
therefore  tends  to  raise  the  movable  plank  E  with  a  force  as 
much  greater,  in  the  aggregate,  than  that  impressed  upon  the 
surface  of  the  smaller,  as  this  surface  is  smaller  than  that  of  the 
larger  column ;  or  (which  is  the  same  thing)  as  the  number  of 
square  inches  in  the  end  of  the  piston  B  is  less  than  that  of  the 
piston  D.  The  power  of  such  a  machine  is  enormously  great ; 
for,  supposing  the  hand  to  be  applied  at  the  end  of  the  handle, 
with  a  force  of  only  ten  pounds,  and  that  this  handle  or  lever  is 
so  constructed  as  to  multiply  that  force  but  five  times,  the  force 
with  which  the  smaller  piston  will  descend  will  be  equal  to  50 
Ibs.  ;  and  let  us  suppose  that  the  head  of  the  large  piston  con- 
tains the  smaller  50  times,  then  the  force  exerted  to  raise  the 
press-board  will  equal  2500  Ibs.  A  man  can  indeed  easily  exert 
many  times  the  force  supposed,  and  can  therefore  exert  a  force 
upon  the  substance  under  pressure,  equal  to  many  tons. 

The  hydrostatic  press  involves  far  less  loss  from  friction  than 
any  other  species  of  press,  and  it  is  said  that  the  naked  force  of 
man  is  more  effective  when  applied  in  this  way  than  in  any 


*  Partington's  Manual  of  Nat.  Phil. 

t  For  a  more  complete  description  of  the  "  Bramah  Press,"  see  Webster's  Principle* 
tf  Hydrostatics,  p.  151. 


264  NATURAL   ."-HILOSOPHY. 

other.  By  the  mere  weight  of  a  man's  body,  when  leaning  on 
the  extremity  of  the  lever,  a  pressure  may  be  produced  of  up 
wards  of  2000  tons.  It  is  the  simplest  and  most  easily  applicable 
of  all  contrivances  for  increasing  human  power  ;  and  the  o*ily 
limit  to  the  force  which  may  be  called  into  action  by  it,  is  the 
want  of  materials  of  sufficient  strength  to  enable  us  to  apply  the 
enormous  pressure  which  it  generates.* 

384.  The  rationale  of  the  principle  of  the  hydrostatic  press 
will  be  best  understood  by  recurring  to  the  following  principles, 
— that  opposite  forces  are  in  equilibrium  when  their  momenta 
are  equal ;  that  a  small  power  may  be  made  to  balance  a  great 
weight,  by  making  it  move  in  a  given  time,  over  a  space  as 
much  greater  than  the  larger  does,  as  its  weight  is  smaller  ;  and 
that  it  may  be  made  to  overcome  that  resistance  or  weight  and 
give  motion  to  it,  if  its  velocity  is  greater  than  that  of  the  latter 
in  a  still  higher  ratio.     Now  to  apply  these  principles  to  the  case 
before  us,  it  is  evident  that  any  quantity  of  water  forced  out  of 
the  smaller  into  the  larger  cylinder,  must  rise  in  the  latter  as 
much  slower  as  the  area  of  the  horizontal  section  is  larger.     If, 
for  example,  the  capacity  of  the  larger  cylinder  were  ten  times 
that  of  the  smaller,  then  a  quantity  of  water  one  inch  in  height 
transferred  from   the  smaller  to  the   greater  cylinder,  would 
occupy  only  the  height  of  one  tenth  of  an  inch,  consequently,  the 
depression  of  the  small  piston  one  inch  would  raise  the  large  one 
only  the  tenth  of  an  inch.     This  case  therefore  resolves  itself 
into  that  general  principle,  according  to  which  a  vast  force  is 
exerted  through  a  short  distance,  by  moving  a  small  force  through 
a  distance  much  greater. 

This  press  is  used  for  the  extraction  of  oils,  either  vegetable 
or  animal,  for  pressing  paper  or  books,  and  for  packing  cotton 
and  other  substances.  Hay  intended  as  food  for  cattle  on  ship- 
board, is  reduced  by  this  press  almost  to  the  state  of  a  solid,  and 
enormous  quantities  are  thus  brought  into  an  inconceivably  small 
compass.  There  would  seem  to  be  no  force  known  to  us  which 
may  not  be  made  to  yield  to  this  power.  It  requires  but  one  of 
its  least  efforts  to  tear  up  a  tree  by  its  roots,  or  to  break  a  large 
beam  asunder. 

385.  The  surface  of  a  fluid  at  rest  is  horizontal. 

The  evidence  of  the  truth  of  this  proposition  is  threefold,  f 
First,  this  result  is  a  natural  consequence  of  the  mobility  of 
fluids,  since,  if  any  portion  is  raised  above  the  rest,  having  no- 
thing to  support  it,  and  being  acted  on  by  gravity,  it  must  de- 

*  Mosely,  Mech.  applied  to  the  Arts,  Art.  197. 

t  It  might  appear  superfluous  to  offer  so  much  proof  of  a  point  so  plain ;  but  the 
several  modes  of  reasoning  will  serve  to  instruct  the  young  learner  in  the  peculiar 
properties  of  fluids. 


HYDROSTATICS.  265 

scend  in  the  same  manner  as  a  body  placed  on  a  perfectly  smooth 
inclined  plane.  Secondly,  whenever  a  body  is  free  to  move,  its 
center  of  gravity  will  descend  as  low  as  possible.  (Art.  70.) 
When,  therefore,  any  portion  of  a  fluid  is  raised  above  the  gene- 
ral level,  the  center  of  gravity  of  the  mass  is  raised,  and  drawn 
out  of  the  line  which  passed  perpendicularly  through  it  and  the 
point  of  suspension,  (Fig.  43,)  and  it  must  return  to  that  line 
before  the  fluid  can  be  at  rest.  Thus,  let  ABCD,  (Fig.  171,)  be 
a  body  of  water,  contained  in  a  cylindrical  vessel ;  let  I  be  the 
center  of  the  surface,  and  IGH  the  perpendicular  line  passing 
through  the  center  of  gravity  G  and  the  base.  Fig.  171.  . 
Through  I,  suppose  a  plane,  movable  on  a  hinge 
at  E,  to  pass,  and  to  be  depressed  into  the  situa- 
tion FIE,  by  which  means  the  water  will  be  de- 
pressed on  the  side  AF  and  raised  on  the  s  e  B 
EB,  while  the  center  of  gravity  will  be  removed  , 
to  the  point  G'.  Now  let  the  plane  be  removed, 
and  the  center  of  gravity  being  free  to  move,  it  D  C 
will  vibrate  around  I  as  a  point  of  suspension,  un-  H 
til  it  finally  recovers  its  situation  at  G,  and  the  surface  of  the 
fluid  will  return  to  its  original  level.  Thirdly,  experience  shows 
that  the  proposition  is  true,  since  fluids,  when  free  to  move,  al- 
ways settle  themselves  with  their  surfaces  parallel  to  the  hori- 
zon.* It  must  be  understood,  however,  that  the  surface  of  large 
bodies  of  water  is  not,  strictly  speaking,  a  horizontal  level,  but 
is  a  portion,  of  the  convex  surface  of  the  earth ;  for  since  the 
center  of  gravity  of  every  portion  of  the  fluid  will  descend  as 
low  as  possible,  the  whole  will  dispose  itself  around  the  center 
of  attraction,  so  as  to  form  a  portion  of  the  earth's  surface.  For 
small  distances  the  curvature  is  so  slight  that  it  may  be  neglect- 
ed, not  amounting  to  one  second  of  a  degree  for  100  feet  ;f  and 
for  the  distance  of  a  mile,  the  deviation  from  a  straight  line, 
drawn  in  the  direction  of  a  tangent,  is  not  more  than  8  inches. 
The  amount  of  the  depressions  for  different  distances,  is  estima- 

2L2 
ted  by  the  following  formula,  D=— '  where   L  represents  the 

number  of  miles,  and  D  the  depression  in  feet.     Thus  the  de- 

2  x4 
pression  of  two  miles  is  — —  =2|  feet.     Let  FE  (Fig.   172,)  or 

*  The  only  exceptions  to  this  law,  are  those  arising  from  the  attraction  of  cohesion 
among  the  particles  of  liquids,  and  the  attraction  exerted  by  small  tubes,  called  capil- 
lary attraction.  In  consequence  of  cohesion,  small  portions  of  a  liquid  form  them- 
selves  into  drops,  and  large  portions  present  a  convex  surface,  as  is  strikingly  exhibit. 
ed  in  a  wine-glass  filled  with  quicksilver.  By  capillary  attraction,  the  surfaces  of 
liquids  are  made  concave,  a  phenomenon  to  which  we  shall  attend  more  particularly 
aereafter.  But  both  these  causes  operate  on  a  scale  comparatively  very  small. 

1 360x60x60x100    1296      9" 
25000X5280     ~132Q~W. 


NATURAL   PHILOSOPHY. 


its  equal  GB,  be  the  depression  for 
the  distance  BE.  For  moderate  dis- 
tances this  arc  may  be  considered  as 
equal  to  its  chord."  Hence  GB(D) : 

EB(L) :  :  EB  :  AB ;  or  D=^g- ,  or  D 

being  expressed  in  feet  and  L  and  AB 

L2x(5280)2    L3X5280 
in  miles,  D= 


Fig.  172. 


ABX5280        7912 


2L8 

— ,  nearly. 

J  mile  -     - 


2  X  I2 


2X31 


=  6  feet. 


do. 


386.  A  practical  application  of  this  principle  is  made  in  the 
art  of  levelling,     A  level  is  sometimes  made  by  merely  cutting 
a  groove  or  channel  in  a  flat  piece  of  board  and  filling  it  with 
water.     When  the  board  is  brought  into  such  a  situation  that 
the  water  in  the  groove  remains  stationary,  the  position  is  hori- 
zontal.    But  the  spirit  level  is  the  instrument  more  commonly 
employed  for  this  purpose.     This  consists  of  a  small  bent  cylin- 
drical tube  of  glass,  from  two  Fig.  173. 

to  six  inches  long,  filled  with  t  ,    ,  _  -^-_-  ________      ,_ 

spirits  of  wine  or  ether,  except 
a  small  space  which  is  occupied 


by  a  movable  bubble  of  air.  When  such  a  tube  is  placed  hori- 
zontally, the  bubble  of  air  will  remain  stationary  in  the  center 
of  the  tube,  at  a  fixed  mark  ;  but  whenever  the  tube  is  inclined, 


Fig.  174. 


in  the  least  degree,  the  bubble  will 
ascend  toward  the  elevated  end. 
Spirit  levels  are  much  used  for  ad- 
justing astronomical,  surveying,  and 
other  delicate  instruments.  Figure 
174  represents  a  levelling  staff  used 
in  surveying  and  grading  lands.  The 
liquid'  in  the  two  arms  of  the  tube 
A  and  B  being  precisely  on  a  level, 
any  two  objects,  P  and  Q,  may  be 
brought  accurately  to  the  same  hori- 
zontal level  by  sighting  P  with  the 
eye  at  A,  and  then  sighting  Q,  with 
the  eye  at  B. 

387.  The  pressure  upon  any  particle  of  a  fluid  of  uniform  densi- 
ty, is  proportioned  to  its  depth  below  the  surface. 


HYDROSTATICS. 


267 


Fig.  175. 


Case  1.  Let  the  column  of  fluid  ABCD, 
(Fig.  175,)  be  perpendicular  to  the  hori- 
zon. Take  any  points,  x  and  y,  at  dif- 
ferent depths,  and  conceive  the  column 
to  be  divided  into  a  number  of  equal 
spaces  by  horizontal  planes.  Then,  since  B  | 
the  density  of  the  fluid  is  uniform 
throughout,  the  pressure  upon  x  and  y, 
respectively,  must  be  in  proportion  to 
the  number  of  equal  spaces  above  them, 
and  consequently  in  proportion  to  their 
depths. 

Case  2.  Let  the  column  be  of  the  same  perpendicular  height 
as  before,  but  inclined  as  is  Fig.  175,  (2) ;  then  its  quantity,  and 
of  course  its  weight,  is  increased  in  the  same  ratio  as  its  length 
exceeds  its  height ;  but  since  the  column  is  partly  supported  by 
the  plane,  like  any  other  heavy  body,  the  force  of  gravity  acting 
upon  it  is  diminished  on  this  account  in  the  same  ratio  as  its  length 
exceeds  its  height ;  therefore  as  much  as  the  pressure  on  the  base 
would  be  augmented  by  the  increased  length  of  the  column,  just 
so  much  it  is  lessened  by  the  action  of  the  inclined  plane  ;  and  the 
pressure  on  any  part  of  Cc  will  be,  as  before,  proportioned  to  its  per- 
pendicular depth ;  and  the  pressure  of  the  inclined  column  ACac 
will  be  the  same  as  that  of  the  perpendicular  column  ABCD. 

There  are  various  experiments  also  by  which  this  proposition 
is  fully  established.* 

388.  According  to  Art.  381,  the  lateral  is  equal  to  the  down- 
ward pressure  ;  and  consequently,  on  this  principle,  may  easily 
be  estimated  the  amount  of  pressure  on  the  sides  of  any  column 
of  water,  or  on  the  banks  of  rivers,  canals,  &c.  At  the  depth  of 
8  feet,  the  pressure  on  a  square  foot  is  equal  to  the  weight  of  a 
column  of  water  whose  base  is  1  foot  and  depth  8  feet,  and  con- 
sequently its  solid  contents  8  cubic  feet ;  and  since  1  cubic  foot 
of  water  weighs  1000  ounces,  or  62£  Ibs.,  therefore  the  weight  of 
the  column =8  x62|=500  Ibs.  Hence  the  pressure  on  a  square 
foot,  at  different  depths,  will  be  as  in  the  following  table  : 


Depth  in  fe 

8 

et.                  Pressure  on  a  square  foot. 

-    -    -     -     500  Ibs. 

Depth  in  feet.                  Pressure  on  a  square  foot. 

56     -     -.-  3500  Ibs. 

16 

-     -     -     -  1000 

64     -     -     -     -  4000 

24 

-     -     -     -  1500 

72     ---     -  4500 

32 

-     -     -     -  2000 

80     -     ---  5000 

40 

...     -  2500 

88     ..--  5500 

48 

-.--  3000 

96     ...     -  6000 

1  mile  or  5280  feet, 

-     -     -     330,000  Ibs. 

544 

1  650  000 

*  Experimental  illustrations  are,  in  this  part  of  the  work,  supposed  to  be  given  by 
the  instructor. 


268  NATURAL   PHILOSOPHY. 

Hence  it  appears  that  at  the  moderate  depth  of  64  feet,  the  pres- 
sure of  a  column  of  water  on  the  bottom  or  sides  of  the  contain- 
ing pipe,  becomes  4000  Ibs.  to  the  square  foot ;  and  the  pressure 
on  the  bottom  of  the  sea,  where  it  is  one  mile  in  depth,  is 
330,000  Ibs.  to  the  square  foot,  and  where  it  is  5  miles  deep,  that 
pressure  is  no  less  than  1,650,000  Ibs.*  From  these  considera- 
tions, wre  may  readily  apprehend  the  cause  of  the  great  difficulty 
experienced  in  confining  a  high  column  of  water ;  and  hence 
also  may  be  inferred  the  immense  pressure  that  is  exerted  on  the 
bottom  of  the  sea.  It  is  said  that  the  Greenland  whale  some- 
times descends  to  the  depth  of  a  mile,  but  always  comes  up  ex- 
hausted, and  blowing  out  blood,  showing  that  the  pressure  had 
so  acted  upon  the  vessels  as  to  cause  them  to  discharge  a  por- 
tion of  their  contents  into  the  lungs,  f 
• 

389.  Indications  of  this  vast  pressure  in  deep  waters,  are  mani- 
fested by  several  interesting  facts.  It  has  long  been  known  to 
mariners,  that  if  a  common  square  bottle  be  let  down  into  the 
sea,  its  sides  are  crushed  inward  before  it  has  reached  the  depth 
of  ten  fathoms.  If  a  stronger  bottle,  (a  common  junk  bottle,  for 
example,)  be  filled  with  water,  corked  close,  and  let  down  to  a 
certain  depth,  either  the  cork  will  be  forced  inward,  or  if  that 
be  secured  in  its  place,  the  salt  water  will  make  its  way  into 
the  bottle  in  spite  of  it,  either  by  compressing  the  cork  or  by 
forcing  in  water  through  it.  It  was  by  sinking  an  apparatus  to 
the  depth  of  500  fathoms,  that  Mr.  Perkins  first  proved  the  com- 
pressibility of  water,  as  mentioned  in  Art.  379.  The  apparatus 
consisted  of  a  hollow  brass  cylinder,  resembling  a  small  cannon,J 
and  furnished  with  a  stopper  so  contrived  as  to  indicate,  when 
the  apparatus  was  drawn  up,  how  far  it  had  been  driven  in 
while  at  the  lowest  depth.  The  same  experiments  were  after- 
wards repeated  on  shore,  a  pressure  being  applied  to  the  plug, 
by  means  of  the  hydrostatic  press,  equivalent  to  2000  atmo- 
spheres. 

The  increase  of  pressure  in  proportion  to  the  depth  of  the  fluid, 
renders  it  necessary  to  make  the  sides  of  pipes  or  masonry,  in 
which  fluids  are  to  be  contained,  stronger  the  deeper  they  go. 
The  same  remark  applies  to  dams,  flood-gates,  and  banks. 

At  the  depth  of  one  mile  the  compression  of  water  is  T3\  ¥  of 
its  bulk,  and  its  specific  gravity  is  increased  in  the  same  ratio ; 
so  that  bodies  which  sink  near  the  surface  of  the  sea,  may  float 
at  a  certain  depth  before  they  reach  the  bottom.  On  the  other 
hand,  a  porous  body,  that  is  light  enough  to  float  near  the  surface, 
will  have  so  much  water  forced  into  its  pores,  when  it  is  sunk  to 
a  great  depth,  as  never  to  rise.  This  is  the  case  with  ships  that 

*  Allowance  must  also  be  made  for  the  saltness  of  the  sea,  salt  water  being  heavier 
than  fresh.  t  Ed.  Phil.  Jour.,  Jan.  1832. 

t  A  cannon  itself  was  afterwards  employed  in  these  experiments. 


HYDROSTATICS. 


are  wrecked  in  deep  water ;  the  parts  of  the  wreck  do  not  rise 
to  the-  surface,  as  they  do  in  shallow  water.* 

390.  The  pressure  of  a  fluid  against  any  surface,  in  a  direction 
perpendicular  to  it,  varies  as  the  area  of  the  surface  multiplied  into 
the  depth  of  its  center  of  gravity  below  the  surface  of  the  fluid. 

When  a  portion,  as  a  square  foot,  of  the  lateral  surface  of  a  col- 
umn of  water,  is  taken,  all  parts  of  it  are  not  equally  distant  from 
the  surface  of  the  fluid  ;  and,  in  this  case,  the  average  depth,  or 
(which  is  the  same  thing)  the  depth  of  the  center  of  gravity,  is 
to  be  understood. 

Let  m,  n,  (Fig.   176,)  be  a  given  Fig-  176- 

portion  of  the  vessel  ABCD,  filled  A 
with  water  or  any  liquid,  and  let  us 
conceive  this  portion  to  be*  occupied 
by  any  number  of  particle^  m,  o,  p, 
n,  &c.  ;  then  the  pressure  produced 
by  all  these  particles  will  be  (Art. 
83)  mXmu-\-o Xox  +p xpy+nxnz,  &c. : 
but  by  a  property  of  the  center  of 
gravity,!  the  sum  of  the  products  is 
equal  to  the  sum  of  the  particles,  that 
is,  the  area  of  the  surface,  multiplied 
into  the  distance  of  the  center  of  grav- 
ity from  the  surface  of  the  fluid.  J  p 

Hence,  the  pressure  on  the  side  of  a  cubical  vessel,  filled  with 
fluid,  is  one  half  the  pressure  against  the  bottom  ;  and  the  whole 
pressure  against  the  sides  and  bottom,  is  equal  to  three  times 
the  weight  of  the  fluid  in  the  vessel.  Fj  177 

K. 

391.  Fluids  rise  to  the  same  level  in  the  opposite 
arms  of  a  recurved  tube. 

Let  ABC,  (Fig.  177,)  be  a  recurved  tube:  if 
water  be  poured  into  one  arm  of  the  tube,  it  will 
rise  to  the  same  height  in  the  other  arm.  For, 
by  Art.  386,  the  pressure  upon  the  lowest  part  at 
B,  in  opposite  directions,  is  proportioned  to  its 
depth  below  the  surface  of  the  fluid.  Therefore, 
these  depths  must  be  equal,  that  is,  the  height  of 
the  two  columns  must  be  equal,  in  order  that  the 
fluid  at  B  may  be  at  rest ;  and  unless  this  part  is 
at  rest,  the  other  parts  of  the  column  cannot  be 
at  rest.  Moreover,  since  the  equilibrium  depends 
on  nothing  else  than  the  heights  of  the  respective 
columns,  therefore,  the  opposite  columns  may  dif- 

*  Arnott. 

t  See  the  last  step  in  the  demonstration  of  Art.  83,  from  which  GKx(p+p+p") 
"  '  'x'+p"Xp"x". 
ncyc.,  '  Hydrodynamics.' 


270  NATURAL   PHILOSOPHY. 

fer  to  any  degree  in  quantity,  shape,  or  inclination  to  the  hori 
zon.  Thus,  if  vessels  and  tubes  very  diverse  in  shape  and  capa- 
city, as  in  Fig.  178,  be  connected  with  a  reservoir,  and  water 
be  poured  into  any  one  of  them,  it  will  rise  to  the  same  level  in 
them  all. 

Fig.  178. 


The  reason  of  this  fact  will  be  further  understood  from  the  ap- 
plication of  the  principle  of  equal  momenta,  (Art.  149  ;)  for  it 
will  be  seen  that  the  velocity  of  the  columns,  when  in  motion, 
win  be  as  much  greater  in  the  smaller  than  in  the  larger  col- 
umns, as  the  quantity  of  matter  is  less ;  and  hence  the  opposite 
momenta  will  be  constantly  equal. 

Hence,  water  conveyed  in  aqueducts  or  running  in  natural 
channels,  will  rise  just  as  high  as  its  source.  Between  the  place 
where  the  water  of  an  aqueduct  is  delivered  and  the  spring,  the 
ground  may  rise  into  hills  and  descend  into  valleys,  and  the  pipes 
which  convey  the  water  may  follow  alHhe  undulations  of  the 
country,  and  the  water  will  run  freely,  provided  no  pipe  is  laid 
higher  than  the  level  of  the  spring.  Waters  running  in  natural 
channels  in  the  earth  are  governed  by  the  same  law. 

392.  The  aqueducts  constructed  by  the  ancient  Romans,  were 
among  the  most  costly  monuments  of  their  arts.  Several  of  them 
were  from  thirty  to  one  hundred  miles  in  length,  and  consisted  of 
vast  covered  canals  built  of  stone.  They  were  carried  over  val- 
leys and  level  tracts  of  country  upon  arcades,  which  were  some- 
times of  stupendous  height  and  solidity.*  From  the  fact  that 
the  ancients  built  aqueducts  with  so  much  labor,  raising  them  to 
a  great  height  in  crossing  valleys,  instead  of  availing  themselves 
of  the  principle  under  consideration,  some  have  supposed  that 
they  were  unacquainted  with  this  principle.  It  appears  never- 
theless that  they  were  acquainted  with  it,  and  even  understood 
the  use  of  pipes  in  conveying  water ;  but  probably  the  expense 
of  pipes,  and  the  difficulty  of  making  them  strong  enough  to  re- 
sist the  pressure  when  laid  at  a  considerable  depth  below  the 
source,  prevented  their  general  use. 

*  Bigelow,  El.  Tech.,  303. 


HYDROSTATICS. 


271 


Fig.  179. 


393.  The  pressure  upon  the  horizontal  base  of  any  vessel  con- 
taining a  fluid,  is  equal  to  the  weight  of  a  column  of  the  fluid, 
found  by  multiplying  the  area  of  the  base  into  the  perpendicular 

height  of  the  column,  whatever  be  the  shape  of  the  vessel. 

This  follows  from  Art.  390,  since  here,,  the 
distance  of  the  center  of  gravity  of  the  base  from 
the  surface  of  the  fluid,  is  the  same  as  the  per- 
pendicular height  of  the  column.  With  a  given 
base  and  height,  therefore,  the  pressure  is  the 
same  whether  the  vessel  is  larger  or  smaller 
above,  whether  its  figure  is  regular  or  irregular, 
whether  it  rises  to  the  given  height  in  a  broad 
open  funnel,  or  is  carried  up  in  a  slender  tube. 
Hence,  any  quantity  of  water,  however  small,  may 
be  made  to  balance  any  quantity,  however  great. 
This  is  called  the  hydrostatic  paradox.  The 
experiment  is  usually  performed  by  means  of 
a  water-bellows,  as  represented  in  Fig.  179. 
When  the  pipe  AD  is  filled  with  water,  the 
pressure  upon  the  surface  of  the  bellows,  and 
consequently  the  force  with  which  it  raises  the 
weights  laid  on  it,  will  be  equal  to  the  weight 
of  a  cylinder  of  water,  whose  base  is  the  surface  of  the  bellows, 
and  height  that  of  the  column  AD.  Therefore,  by  making  the 
tube  small,  and  the  bellows  large,  the  power  of  a  given  quantity 
of  water,  however  small,  may  be  increased  indefinitely.  The 
pressure  of  the  column  of  water  in  this  case  corresponds  to  the 
force  applied  by  the  piston  in  the  hydrostatic  press,  (Art.  381,) 
and  the  explanation  according  to  the  principle  of  equal  momenta, 
is  the  same  in  both  cases.* 

394.  The  principle  of  the  Hydrostatic  Paradox,  is  sometimes 
exemplified  by  pouring  liquids  into  casks,  through  long  tubes  in- 
serted in  the  bung  holes.     As  soon  as  the  cask  is  full,  and  the 
water  rises  in  the  pipe  to  a  certain  height,  the  cask  bursts  with 
violence.     The  same  cause  is  supposed  sometimes  to  produce 
great  effects  in  nature,  such  as  splitting  rocks,  heaving  up  moun- 
tains, and  other  effects  resembling  earthquakes.     For,  suppose 
that  in  the  interior  of  a  mountain  there  were  an  empty  space  ten 
yards  square,  and  only  an  inch  deep,  in  which  the  water  had 
lodged  so  as  to  fill  it  entirely  ;  and  suppose  that  a  crevice  in  the 
earth  should  extend  from  this  spot  two  hundred  feet  above,  which 
should  also  become  filled  with  water  by  rain  or  otherwise :  the 


*  The  bellows  rises  through  so  small  a  space,  that  its  motion  is  hardly  perceptible, 
out  it  may  be  rendered  very  striking  by  connecting  with  the  bellows  (as  is  done  in 
the  lecture  room  at  Yale  College)  a  lever,  and  several  multiplying  wheels,  whjch 
give  a  rapid  motion  to  a  pointer. 


JB72  NATURAL   PHILOSOPHY. 

force  exerted  would  be  adequate  to  shake  the  mountain  and  per- 
haps rend  it  asunder.* 

395.  Although  the  weight  of  a  given  quantity  of  water  would 
not  be  altered  by  varying  the  shape  of  the  vessel,  yet  the  pressure 
which  it  exerts  on  the  bottom  of  the  vessel  will  be  greater  in 
proportion  as  the  altitude  of  the  mass  is  greater,  and  of  course 
greater  in  a  harrow  vessel  than  in  a  wide  one.     If  it  be  asked 
why  the  weight  is  not  increased  as  the  downward  pressure  is 
increased,  the  answer  is,  that  the  pressure  in  that  direction  is 
exactly  counterbalanced  by  an  equal  pressure  in  the  opposite 
direction. 

OP    SPECIFIC    GRAVITY. 

396.  The  Specific  Gravity  of  a  body,  is  its  weight  compared  with 
the  weight  of  another  body  of  the  same  bulk,  taken  as  a  standard. 

Water  is«  the  standard  for  all  solids  and  liquids,  and  common 
air  for  gases.  Therefore  the  specific  gravity  of  a  solid  or  a 
liquid  body,  is  the  ratio  of  its  weight  to  the  weight  of  an  equal 
volume  of  water ;  and  the  specific  gravity  of  an  aeriform  body, 
is  the  ratio  of  its  weight  to  the  weight  of  an  equal  volume  ol 
air.  But  a  ratio  is  expressed  by  a  vulgar  fraction,  whose  nume- 
rator is  the  antecedent,  and  whose  denominator  is  the  consequent. 
If,  therefore,  the  weight  of  a  body  is  made  the  numerator,  and 
the  weight  of  an  equal  volume  of  water  the  denominator,  the 
value  of  the  fraction,  that  is,  the  quotient,  will  express  the  specific 
gravity  of  the  body.  Hence,  the  weight  of  a  body  being  given, 
and  being  made  the  numerator,  every  process  for  finding  the  spe- 
cific gravity,  consists  in  finding  for  the  denominator  the  weight 
of  an  equal  bulk  of  water  or  air.  The  principles  upon  which 
the  methods  of  doing  this  depend,  are  now  to  be  explained. 

397.  A  body  immersed  in  a  fluid,  loses  as  much  weight,  as  is 
equal  to  the  weight  of  an  equal  volume  of  the  fluid. 

Let  EF,  (Fig.    180,)  be  a  solid  body  Fig.  180. 

immersed  in  a  vessel  of  water  or  any 
fluid,  and  suppose  it  divided  into  an  in- 
definite number  of  perpendicular  col- 
umns reaching  to  the  surface  of  the  flu- 
id, as  mon.  Now  the  upward  pressure 
at  n  is  as  its  depth,  and  the  downward 
pressure  at  o  as  its  depth ;  therefore  the 
upward  pressure  exceeds  the  downward, 
by  the  weight  of  a  column  of  water  equal  to  no.  The  same  is 
true  of  all  the  columns,  however  numerous  they  may  be,  that 

*  Library  of  Useful  Knowledge,  '  Hydrostatics.' 


HYDROSTATICS.  273 

can  be  drawn  parallel  to  no ;  but  these  columns,  taken  collec- 
tively, make  up  a  body  of  water  equal  in  bulk  to  the  solid. 
Hence  the  solid  is  pressed  upward  more  than  downward,  by 
the  weight  of  a  quantity  of  water  of  the  same  magnitude,  and 
consequently  loses  so  much  of  its  weight.  Hence  the  specific 
gravity  of  any  solid  body  that  will  sink  in  water,  is  found  by 
the  following 

RULE. — Divide  the  weight  of  the  body  by  its  loss  of  weight  in 
water. 

398.  When  the  body  whose  specific  gravity  is  required  is 
lighter  than  water,  as  a  cork,  for  example,  the  object  is  still  to 
find  the  weight  of  an  equal  bulk  of  water,  since  that  will  con- 
stitute the  denominator,  or  divisor,  as  before.     To  ascertain  this, 
suspend  any  heavy  body,  as  a  mass  of  lead  or  glass,  in  water, 
and  find  its  weight.     Attach  to  it  the  lighter  body.     Now  the 
cork  will  not  only  lose  its  own  weight,  but  will  diminish  the 
weight  of  the  heavy  body  ;  and  the  weight  of  an  equal  bulk  of 
water  will  be  indicated  by  the  whole  of  what  the  cork  loses, 
namely,  its  own  weight  added  to  the  loss  occasioned  to  the  other 
body.     Whence  we  have  the  following 

RULE. — To  find  the  specific  gravity  of  a  body  lighter  than 
water,  Divide  its  weight  by  the  sum  of  its  weight  added  to  the  loss 
of  weight  which  it  occasions  in  a  heavy  body  previously  balanced  in 
water. 

399.  A  solid  which  is  soluble  in  water,  as  a  lump  of  salt,  is 
protected  from  solution  by  covering  it  with  oil  or  a  thin  coat  of 
beeswax :  and  solids  that  are  very  porous  and  would  absorb 
water,  and  thus  increase  their  specific  gravities,  as  certain  kinds 
of  wood,  are  first  covered  with  varnish.*     The  specific  gravity 
of  solid  substances,  which  are  too  minutely  divided  to  be  weighed 
in  water,  separately,  as  grains  of  sand  or  shot,  may  be  found  by 
weighing  them  in- a  small  bucket  previously  balanced  in  water 

400.  The  specific  gravity  of  liquids  may  be  ascertained  bj 
several  different  methods. 

RULE  I. — Weigh  equal  volumes  of  the  liquid  and  of  water,  and 
divide  the  former  result  by  the  latter. 

RULE  II. — Ascertain  the  loss  of  iceight  of  any  solid  body  first  in 
the  liquid  and  then  in  water,  and  divide  the  former  result  by  the 
latter. 

Both  these  rules  obviously  depend  upon  the  same  principles 
as  those  explained  in  Art.  396,  the  weight  of  the  liquid  being 
immediately  compared  with  that  of  an  equal  bulk  of  water ;  but 
there  is  another  method,  founded  on  the  following  proposition. 

*  Cavallo,  I,  212. 
35 


274 


NATURAL    PHILOSOPHY. 


401.  Two  columns  of  fluids  of  different  specific  gravities,  press 
ing  freely  on  each  other  at  their  bases,  balance  one  another  when 
their  heights  are  inversely  as  their  specific  gravities. 

Let  AB,  (Fig.   181,)  be  a  recurved  tube,  and  let     Fig.  181. 
the  height  of  the  column  of  the  fluid  B  be  as  much 
greater  than  that  of  A,  as  the  fluid  B  is  lighter  than 
the  fluid  A ;  the  two  columns  will  then  be  in  equi- 
librio. 

If  the  tube  be  of  uniform  bore  throughout,  then 
the  proposition  is  manifestly  true,  because  the  quan- 
tities of  matter  pressing  on  each  other  in  opposite 
directions  will  be  equal  and  will  have  equal  mo- 
menta ;  but  from  the  peculiar  nature  of  fluids,  (Art. 
391,)  the  opposite  pressures  will  be  the  same,  when 
the  heights  of  the  columns  are  the  same,  whatever 
may  be  the  shape  or  capacity  of  the  tube.  If  we 
introduce  mercury  into  one  arm  of  the  tube  and 
water  into  the  other,  the  graduated  scale  will  indi- 
cate that  the  water  stands  13£  times  as  high  as  the  A; 
mercury.  Therefore  the  specific  gravity  of  mercury 
is  13£.  Proof  spirit  will  stand  at  .923  ;  sweet  oil  at 
.915;  and  their  specific  gravities  are  the  same, 
water  being  1. 

402.  These  comprehend  the  most  common  methods 

of  determining  the  specific  gravities  of  bodies  ;  but  when  great 
accuracy  is  required,  the  process  of  finding  the  specific  gravity 
becomes  one  of  much  delicacy.  The  balance  must  be  very 
sensible  ;  the  body,  when  suspended  in  water,  must  hang  by  a 
fine  thread  or  hair ;  and  the  water  must  be  of  a  standard  tem- 
perature, since  it  alters  its  density  by  a  change  of  temperature, 
so  that  the  quantity  of  water  which  we  wish  to  find,  (namely,  a 
quantity  equal  in  magnitude  to  the  given  body,)  will  weigh  more 
when  the  water  is  cold  and  less  when  it  is  warm.* 

As  expeditious  methods  of  finding  the  specific  gravity  are 
sought  for  in  commerce  and  the  arts,  various  instruments  have 
been  invented  for  this  purpose,  which  are  called  in  general  hy- 
drometers. But  before  we  can  understand  the  principles  on 
which  these  instruments  depend,  we  must  attend  to  the  theory 
of  floating  bodies,  so  far  as  it  is  contained  in  the  following  propo- 
sitions. 

403.  If  a  body  floats  on  a  fluid,  it  displaces  as  much  of  the  fluid 
as  is  equal  to  its  own  weight. 

*  The  standard  temperature  is  not  universally  agreed  on.  The  temperature  of  60° 
is  usually  understood,  at  which  a  cubic  foot  of  water  weighs  62.353  Ibs.  or  a  little  less 
than  1000  ounces.  A  better  standard  is  that  of  40°,  at  which  temperature  the  weight 
of  a  cubic  foot  is  still  nearer  to  62J  Ibs.  or  '000  ounces. 


HYDROSTATICS.  275 

For  the  body  P+Q*  is  supported  by  the  pressure  of  the  fluid, 
upward  against  the  part  immersed.  But  before  the  body  was 
immersed,  the  same  pressure  supported  a  quantity  of  the  fluid 
which  occupied  the  same  space  and  was  therefore  of  the  same 
magnitude  with  Q ;  and  since  the  same  pressure  must  sustain 
the  same  weight,  when  there  is  an  equilibrium,  the  weight  of 
the  body  must  be  equal  to  the  weight  of  a  quantity  of  the  fluid 
equal  in  bulk  to  Q. 

This  proposition  is  also  easily  established  by  experiment ;  for 
if  into  a  vessel  full  of  water  a  floating  body,  as  a  piece  of  wood, 
be  introduced,  the  quantity  of  water  displaced  will  be  found  to 
be  exactly  equal  in  weight  to  the  body.  Or  if  the  vessel  full  of 
water  be  accurately  balanced  in  a  scale,  and  then  removed  and 
the  piece  of  wood  introduced,  on  restoring  the  vessel  to  the  scale 
it  will  still  remain  in  equilibrium,  the  wood  exactly  compensating 
for  the  water  it  displaced. 

404.  If  a  body  floats  on  a  fluid,  the  part  immersed  bears  the  same 
ratio  to  the  whole  body,  as  the  specific  gravity  of  the  body  bears  to 
that  of  the  fluid. 

Let  S  be  the  specific  gravity  of  the  fluid,  and  s  that  of  the 
solid ;  then  the  absolute  weight  of  the  body  (P+Q)*  will  equal 
(P+Q)  xs,f  and  the  weight  of  a  quantity  of  the  fluid  equal  to 
Q  is  QxS;  and  by  Art.  403,  these  weights  are  equal  to  each 
other,  that  is,  (P+Q)  xs=QxS  .-.  Q  :  P+Q  : :  s  :  S. 

405.  A  floating  body  will  be  at  rest,  only  when  its  center  of  gravity 
is  in  the  same  vertical  line  with  the  center  of  gravity  of  the  part  of 
the  fluid  displaced. 

As  the  space  from  which  the  fluid  has  been  displaced,  exactly 
copies  the  figure  of  the  part  of  the  body  immersed,  the  center 
of  gravity  of  the  displaced  fluid  is  the  same  with  that  belonging 
to  the  figure  of  the  segment.  Let  this  center  of  gravity  be  at  C, 
(Fig.  182,)  while  that  of  the  entire  body  is  at  G.  Now  the  fluid, 


Fig.  182. 


previous  to  its  removal,  was  sustained  by  an  upward  force  equal 

*  Q  represents  the  part  immersed  ;  P  the  other  part. 

W 
t  *  denotes  the  specific  gravity  of  the  body,  water  being  1 ;  and  s=  •       -,   (Art. 

396,).-.W=(P+Q)x*. 


276  NATURAL   PHILOSOPHY. 

to  its  own  weight,  acting  in  the  vertical  line  DC ;  and  the  same 
upward  force  now  acts  upon  the  floating  body  at  the  point  C. 
But  the  body,  being  free  to  move,  is  carried  downward  by  a 
force  acting  in  the  direction  of  the  vertical  line  GH.  Were 
these  two  forces  exactly  opposite  and  equal,  they  would  keep 
the  body  at  rest ;  but  this  is  the  case  only  when  the  points  C  and 
G  are  in  the  same  vertical  line  :  in  every  other  position  of  these 
points  the  two  parallel  forces  tend  to  turn  the  body  round.  (Art. 
70.) 

When  the  floating  body  is  a  regular  solid,  as  a  sphere,  the 
space  occupied  by  the  immersed  segment  being  always  the  same, 
its  center  of  gravity  is  immovable.  If  the  body  is  of  uniform 
density,  so  that  the  center  of  gravity  coincides  with  the  center  of 
magnitude,  it  will  remain  at  rest  in  every  position ;  for  the 
center  of  gravity  of  the  whole  body  and  of  the  segment,  will 
always  be  in  the  same  vertical  diameter.  But  if  the  sphere  is 
of  unequal  density,  then  it  will  turn  around  the  center  of  gravity 
of  the  segment  as  around  a  fixed  point,  and  be  at  rest  only  when 
its  center  of  gravity  is  either  directly  above  or  directly  below 
that  of  the  segment. 

Persons  have  sometimes  attempted  to  walk  on  the  water,  by  at- 
taching to  the  feet  inflated  bladders  or  air  bags  of  India  rubber. 
The  body  can  be  kept  from  reversing  its  position,  only  by  main- 
taining its  center  of  gravity  directly  over  that  of  the  fluid  dis- 
placed,— a  point  of  such  delicacy,  as  to  require  great  skill  and 
dexterity.  The  feat  is  rendered  easier  by  carrying  a  staff  having 
a  blown  bladder  at  the  end  of  it,  by  which  three  points  of  sup- 
port are  commanded,  and  consequently  a  greater  breadth  of  base 
is  secured.  Most  persons,  however,  in  attempting  this  feat, 
would  shortly  find  their  heads  downward,  and  their  feet  out  of 
water. 

Life  preservers,  consisting  of  air  bags  attached  to  the  waist, 
act  on  the  foregoing  principles,  and,  in  numerous  instances,  have 
been  instrumental  in  saving  the  lives  of  shipwrecked  mariners. 

405'.  When  a  floating  body  rolls  on  one  side,  so  as  slightly  to 
disturb  the  center  of  gravity  of  the  fluid  displaced,  (or  that  which 
occupied  the  space  now  occupied  by  the  body,)  the  point  of  in- 
tersection of  the  vertical  through  the  center  of  gravity  of  the 
fluid  displaced,  with  the  vertical  through  the  center  of  gravity 
of  the  body  when  at  rest,  is  called  the  Metacenter.  Let  Fig.  183 
represent  a  vessel  turned  a  little  on  one  side ;  and  let  G  be  the 
center  of  gravity  of  the  vessel,  AB  the  surface  of  the  water,  and 
C  the  center  of  gravity  of  the  fluid  displaced.  When  the  vessel 
floated  at  rest,  GC  was  vertical ;  but  when  it  rolled  slightly  on 
one  side,  GC  becomes  inclined  to  the  vertical  in  a  small  angle, 
and  the  center  of  gravity  of  the  fluid  displaced,  shifts  its  position 
to  C'.  Let  the  vertical  through  C'  meet  GC  in  M,  then  M  is  the 


HYDROSTATICS. 

Fig.  183. 


277 


Metacenter.  Now  the  weight  of  the  vessel  acts  downward 
through  G,  while  the  upward  pressure  of  the  water  acts  upward 
in  the  direction  of  C'M,  and  both  forces  tend  to  right  the  vessel, 
or  to  bring  it  back  to  its  original  position.  Hence,  when  the 
Metacenter  is  above  the  center  of  gravity  of  the  vessel,  the 
equilibrium  is  stable,  and  the  vessel  readily  and  forcibly  recovers 
its  fixed  position  But  suppose  M  and  G  to  change  places,  then 
these  two  forces  will  conspire  to  turn  the  vessel  over  still  more, 
and  the  equilibrium  is  unstable.  If  the  Metacenter  coincides 
with  the  center  of  gravity,  the  equilibrium  is  stable  in  any  posi- 
tion of  the  vessel.  Balloons  in  rising  from  the  ground  forcibly 
right  themselves  in  accordance  with  the  foregoing  principle. 

Hence  the  necessity  of  having  not  only  the  heavier  parts  of  a 
ship's  cargo  stowed  at  the  bottom  of  the  vessel,  but  also  of  hav- 
ing the  vessel  ballasted,  or  the  keel  heavily  laden,  is  apparent. 
For  the  masts  and  rigging  may  raise  the  center  of  gravity  of  the 
vessel  above  the  center  of  gravity  of  the  water  displaced,  in 
which  case  the  vessel  will  be  very  liable  to  upset.  The  danger 
from  the  rolling  of  large  vessels  arises  from  the  liability  of  parts 
of  the  cargo  to  shift,  in  which  case  the  equilibrium  may  cease 
to  be  stable  ;  and  the  danger  of  standing  up  in  a  small  boat  is 
quite  apparent,  for  the  elevation  of  the  body  will  certainly  raise 
the  center  of  gravity  of  the  floating  mass  above  the  center  of 
gravity  of  the  water  displaced.*  ^ 

406.  The  HYDROMETER,  an  instrument  used  in  commerce  for 
determining  at  once  the  specific  gravity  of  liquors,  depends  on 
the  principle  enunciated  in  Art.  404. 

The  most  common  hydrometer  is  represented  in  Fig.  184.  It 
consists  of  a  hollow  ball,  with  a  long  slender  stem,  and  since  it 
will  sink  until  it  has  displaced  a  quantity  of  the  fluid  equal  in 
weight  to  itself,  it  will  of  course  sink  to  a  greater  depth  the 
lighter  the  fluid.  From  the  depths  to  which  it  sinks,  therefore, 
as  indicated  by  the  graduated  stem,  the  corresponding  specific 
gravities  are  estimated  according  to  the  formula  in  Art.  404,  and 

*  Webster's  "  Principles  of  Hydrostatics"  p.  59. 


278 


NATURAL   PHILOSOPHY. 


Fig.  184. 


arranged  in  a  table.  Nicholson's  Hydrometer,  (Fig.  185,)  is  the 
most  useful  of  this  class  of  instruments,  since  it  may  be  applied 
to  finding  the  specific  gravities  of  solid  bodies.  In  addition  to 
the  hollow  ball  of  the  common  hydrometer,  it  is  furnished  with 
a  dish  AB  for  receiving  weights,  and  a  stirrup  EF  for  holding  the 
substance  under  trial.  The  instrument  is  so  adjusted  that  when 
1000  grains  are  placed  in  the  upper  dish,  the  whole  will  sink  in 
distilled  water  at  the  temperature  of  60°  Fahr.  to  the  point  m  in 
the  middle  of  the  stem.  If,  therefore,  in  the  case  of  different 
fluids,  we  add  weights  until  m  stands  at  the  level  of  the  fluids, 
the  quantities  displaced  will  be  the  same  in  all,  and  the  weight 
of  each  quantity  will  be  that  of  the  instrument  W  together  with 
the  weight  w  added  to  the  dish,  which  will  constitute  the  nume- 
rator of  the  fraction  expressing  the  specific  gravity,  (Art.  396,) 
while  W+1000  expresses  the  denominator,  being  the  weight  of 
an  equal  bulk  of  water  ;  and  since  the  weight  of  the  instrument 
is  known,  the  only  thing  to  be  determined  in  any  case  is  the 
weight  to  be  added,  in  order  to  sink  the  point  m  to  the  level  of 


the  fluid.     Then 


W+MJ 


—  =the  specific  gravity. 


W+1000 

To  determine  the  specific  gravity  of  a  solid  body  not  exceed- 
ing 1000  grains  in  weight,  we  first  place  the  instrument  in  water, 
put  the  body  into  the  dish,  and  add  weights  to  sink  the  point  m 
to  the  level  of  the  water.  Then,  since  the  weight  of  the  body 
itself  and  the  weight  added,  together  equal  1000  grains,  the 


HYDROSTATICS.  279 

weight  of  the  body=1000  —  the  weight  added.  Now  we  transfer 
the  body  to  the  stirrup  EF,  and  ascertain  what  weight  must  be 
added  to  the  dish  to  sink  it  to  the  same  level  as  before.  This 
weight  will  show  the  loss  of  weight  of  the  body  in  water  ;  and 
the  weight  in  air,  divided  by  the  loss  of  weight  in  water,  gives 
the  specific  gravity.  As  the  cylindrical  stem  of  the  instrument 
is  only  one  fortieth  of  an  inch  in  diameter,  the  instrument  will 
rise  or  fall  nearly  one  inch  by  the  subtraction  or  addition  of  one 
tenth  of  a  grain.  It  will  therefore  indicate  a  change  of  weight 
less  than  one  twentieth  of  a  grain,  and  give  the  specific  gravity 
correct  to  five  places  of  decimals.* 

407.  An  accurate  knowledge  of  the  specific  gravities  of  bodies 
is  of  great  use  for  many  purposes  of  science  and  the  arts,  and 
they  have  therefore  been  determined  with  the  greatest  possible 
precision.  The  heaviest  of  all  known  substances  is  platina, 
whose  specific  gravity,  in  its  state  of  greatest  condensation,  is 
22,f  water  being  1  ;  and  the  lightest  of  all  ponderable  bodies  is 
hydrogen  gas,  whose  specific  gravity  is  .073,  common  air  being 
1.  Also,  air  is  818  times  lighter  than  water.  Hence,  by  calcu- 
lation, it  will  be  found  that  platina  is  about  247,000  times  as 
heavy  as  hydrogen,  and  a  wide  range  is  allowed  to  the  various 
bodies  which  lie  between  these  extremes.  The  metals,  as  a 
class,  are  the  heaviest  bodies  ;  next  to  these  come  the  metallic 
ores  ;  then  the  precious  gems  ;  and  finally,  minerals  in  general, 
animal,  liquid,  and  vegetable  substances,  in  order,  according  to 
the  following  table. 
Metals,  (pure,)  not  including  the  bases  of  the  alkalies  and 

earths,  from       ........    5  to  22 

Gold,  19.25  Steel,    7.84 

Quicksilver,  13.58  Iron,     7.78 

Lead,  11.35  Tin,      7.29 

Silver,  10.47  Zinc,     7.00 

Copper,  8.90 

Metallic  Ores,  lighter  than  the  pure  metals,  but  usually 

above  4.005. 

Precious  Gems,  as  the  ruby,  sapphire,  and  diamond,  3  —  4 

Minerals,  comprehending  most  stony  bodies,     -         -  2  —  3 

Liquids,  from  ether  highly  rectified  to  sulphuric  acid  highly 

concentrated,        ......_  |  —  2§ 

Acids  in  general,  heavier  than  water. 

Oils          do.  lighter;  but  the  oils  of  cloves  and  cinna- 

mon are  heavier  than  water  ;  the  greater  part  lie  be- 

tween .9  and  1.          --..._.       .9  —  \ 


Ed.  Encyc.  X,  781.  f  Ib.  794. 

In  a  few  instances,  metallic  ores,  when  largely  combined  with  foreign  ingredients, 

below  3.  §  Ed.  Encyc. 


. 

t  In  a  few  i 
fall 


280  NATURAL   PHILOSOPHY. 

Milk, 1.032 

Alcohol,  (perfectly  pure,) .797 

Do.      of  commerce, .835 

ProofSpirit, .923 

Wines  ;  the  specific  gravity  of  the  lighter  wines,  as  Cham- 
pagne and  Burgundy,  is  a  little  less,  and  of  the  heavier 
wines,  as  Malaga,  a  Ijttle  greater  than  that  of  water. 
Woods,  cork  being   the  lightest,  and   lignum  vitae    the 
heaviest,  -        -.  -        -        -        -        -          £  to  li* 

408.  If  we  balance,  in  a  pair  of  scales,  a  tumbler  filled  with 
water,  to  a  certain  mark  near  the  top,  and  then,  turning  out  all 
the  water  except  a  small  quantity,  introduce  any  solid  body,  (as 
a  tumbler  a  little  less  than  the  first,)  so  as  to  raise  the  water  on 
the  sides  to  the  same  mark  as-  before,  the  equilibrium  will  be 
restored.     Here,  the  space  occupied  by  the  solid  immersed,  is 
the  same  with  that  before  occupied  by  the  water.     On  the  same 
principle,  a  ship  is  floated  in  a  dock  with  a  very  small  quantity 
of  water,  and  still  rides,  as  freely  as  on  the  ocean.     By  the  ascent 
of  the  water  on  the  sides,  the  upward  pressure  on  the  bottom  is 
increased,  on  the  same  principle  as  in  the  Hydrostatic  Paradox, 
(Art.  393.)     Though,  in  this  case,  we  cannot  say  that  a  quantity 
of  water  is  displaced  equal  in  weight  to  the  solid,  (since  the  whole 
of  the  water  originally  in  the  dock  may  not  have  been  nearly  suf- 
ficient to  fill  the  space  occupied  by  the  ship,)  yet  the  effect  is  the 
same,  in  regard  to  the  pressure  on  the  water  below  the  ship,  and 
of  course  on  the  upward  pressure,  (Art.  381,)  as  though  the  space 
occupied  by  the  ship  below  the  level  of  the  fluid  on  its  sides,  were 
filled  with  water.     On  this  principle,  the  weight  of  a  loaded  boat 
in  the  lock  of  a  canal  is  easily  estimated. 

Boats  are  sometimes  made  of  iron  instead  of  wood,  their  thick- 
ness being  so  much  less,  that  the  entire  weight  of  the  boat  is  not 
greater  than  when  made  of  wood. 

409.  The  human  body,  when  the  lungs  are  filled  with  air,  is 
lighter  than  water,  and  but  for  the  difficulty  of  keeping  the  lungs 
constantly  inflated,  it  would  naturally  float,  f     With  a  moderate 
degree  of  skill,  therefore,  swimming  becomes  a  very  easy  pro- 
cess, especially  in  salt  water.     When,  however,  a  man  plunges, 
as  divers  sometimes  do,  to  a  great  depth,  the  air  in  the  lungs  be- 
comes compressed,  and  the  body  does  not  rise  except  by  muscular 
effort.     The  bodies  of  drowned  persons  rise  and  float  after  a  few 
days,  in  consequence  of  the  inflation  occasioned  by  putrefaction. 

*  The  lightest  kinds  of  wood,  are  poplar,  pear,  and  sassafras,  all  of  which  are  be- 
low .5 ;  the  heaviest  are  ebony,  cedar,  mahogany,  oak-heart,  box,  and  pomegranate, 
which  are  heavier  than  water.  Wood  increases  in  density  by  age.  Knots  have 
been  known  to  reach  the  specific  gravity  of  1.76. 

t  See  Dr.  Franklin's  remarks  ou  this  subject,  in  his  Works. 


HYDROSTATICS.  281 

Quadrupeds  swim  much  more  easily  than  man,  because  the  mo- 
tions of  the  limbs  necessary  to  sustain  themselves,  nearly  coin- 
cide with  their  natural  motions  in  walking,  while  the  body  main- 
tains nearly  its  usual  posture. 

410.  If  a  body  is  held  beneath  the  surface  of  a  fluid,  the  force 
with  which  it  will  ascend,  if  it  is  lighter  than  the  fluid,  or  with  which 
it  will  descend,  if  it  is  heavier,  is  equal  to  the  difference  between  its 
own  weight  and  the  weight  of  an  equal  quantity  of  the  fluid. 

Let  s  be  the  specific  gravity  of  the  solid  and  S  that  of  the  fluid, 
and  M  the  weight  of  an  equal  bulk  of  water.  Now  the  body  held 
beneath  the  water  obviously  descends  with  its  own  weignt=M 
xs*  while  it  is  pressed  upward  with  the  weight  of  the  quantity 
of  fluid  displaced=MxS  ;  consequently,  the  force  with  which  it 
ascends  must  be  (MxS)  —  (Mxs),  and  the  force  with  which  it 
descends  is  (M  xs)  —  M  x  S,  which  are  the  differences  between  the 
weight  of  the  body  and  the  weight  of  the  fluid  displaced.! 

411.  On  the  foregoing  principle  is  founded  the  construction  of 
a  machine  called  the  Camel,  for  raising  sunken  vessels,  or  for  lift- 
ing ships  over  sand  banks.    A  similar  effect  is  exhibited  in  rivers, 
where  the  ice  is  formed  upon  the  stones  at  their  bottom.     Ice  is 
specifically  lighter  than  water,  and  therefore  when  it  accumulates 
to  a  certain  degree  round  the  stones,  the  upward  pressure  upon 
the  stones  exceeds  their  pressure  downward,  and  they  are  brought 
to  the  surface,  having  been  sometimes  torn  up  with  great  force. 
Huge  masses  of  stones  appear  in  many  cases  to  have  been  floated 
by  the  ice  adhering  to  them,  and  carried  to  a  great  distance  from 
the  place  of  their  formation.  J 

412.  Rocks  and  stones  being  only  a  little  more  than  twice  as 
heavy  as  water,  of  course  nearly  half  their  weight  is  sustained 
while  they  are  immersed  in  water ;  and  hence  the  increased  weight 
which  is  felt  when  a  large  stone  is  lifted  from  the  bed  of  a  river, 
as  soon  as  it  reaches  the  surface.     Large  masses  of  rocks  are 
transported  with  far  greater  facility  by  torrents,  on  account  of 
their  diminished  weight.     On  the  same  principle,  the  limbs  feel 
very  heavy  after  lying  for  some  time  in  a  bath.     Life-boats  have 
a  large  quantity  of  cork  mixed  in  their  structure  ;  or  of  air-tight 
vessels  of  thin  copper  or  tin  plate,  so  that,  even  when  the  boats 
are  filled  with  water,  a  considerable  part  still  floats  above  the 
surface.§ 

When  light  bodies  are  attached  to  heavier  for  the  purpose  of 
making  them  float,  some  caution  is  necessary,  since,  on  account 
of  the  tendency  of  the  center  of  gravity  to  seek  the  lowest  point, 

«  For  s=^-.-.W=MXs.  t  Ed.  Encyc.  X,  722.          t  Ib.  §  Arnott 


282  NATURAL   PHILOSOPHY. 

there  will  be  danger  of  upsetting.  Thus,  persons  endeavoring  to 
swim  by  attaching  to  their  persons  blown  bladders,  sometimes 
have  their  heads  turned  downward ;  and  a  man  undertaking  to 
walk  on  the  water  in  a  pair  of  cork  boots,  would  shortly  disap- 
pear, and  nothing  would  be  seen  except  the  boots  sticking  out  of 
the  water.*  (Art.  405.) 

413.  The  magnitude  of   bodies    may  often  be    most  conve- 
niently and  accurately  estimated  from  the  doctrine  of  specific 
gravities.     Suppose  we  wish  to  ascertain  the  exact  number  of 
solid  inches  contained  in  a  stone  of  rude  and  irregular  shape,  we 
should  find  great  difficulty  in  applying  to  it;  any  linear  measure- 
ment ;  but  if  we  ascertain  its  loss  of  weight  in  water,  then  we 
have  the  weight  of  an  equal  bulk  of  water,  and  since  1000  ounces 
contain  1728  cubic  inches,  we  may  easily  find  how  many  cubic 
inches  correspond  to  the  weight  of  water  of  equal  magnitude 
with  the  body  in  question. 

We  may  estimate  the  quantity  of  ice  in  an  island  of  ice,  such 
as  occur  in  the  northern  seas,  by  applying  the  principle  demon- 
strated in  Art.  404. 

414.  As  examples  of  the  manner  in  which  questions  are  solved 
by  the  principles  of  Hydrostatics,  we  subjoin  a  few  problems  of 
the  more  difficult  class,  with  their  solutions,  to  which  we  shall 
add  a  variety  of  questions  as  an  exercise  for  the  learner. 


1.  What  is  the  weight  of  a  chain  of  pure  gold,  which  raises 
the  water  1  inch  in  height,  in  a  cubical  vessel  whose  side  is  3 
inches  ?  and  suppose  a  chain  of  the  same  weight  were  adulte- 
rated with  14^  oz.  of  silver,  how  much  higher  would  it  raise  the 
water  in  the  vessel  ? 

The  contents  of  the  chain  are  3x3x1  =9  cubic  inches. 
Since  a  cubic  foot  or  1728  inches  of  distilled  water  weigh  1000 
oz.f  avoirdupois,  and  since  192  oz.  avoirdupois=175  troy,  in 
which  gold  and  silver  are  weighed;  therefore,  1728  inches  of 
water  weigh  911.458  oz.  troy ;  or  1  cubic  inch  weighs  .5274  oz. 
The  specific  gravity  of  gold  being  19.25,  and  that  of  silver  being 
10.47,  therefore  1  inch  of  gold  weighs  10.15  oz.  and  1  inch  of 
silver  weighs  5.52  oz.  Hence  the  chain  weighs  91.35  oz.  Again, 
10.15  oz.  :  1  in. :  :  14.5  oz.  :  1.428=cub.  inch,  of  gold  in  14.5  oz. 
5.52  :  1  :  :  14.5  :  2.626=  do.  silver  do. 

.-.  9— 1.428+2.626  =  10.198=  column  of  water  raised  by  the 

*  Arnott. 

t  By  act  of  Parliament,  1825,  19  cubic  inches  of  distilled  water,  temp.  50°  Fahr., 
weigh  10  oz.  troy.  These  numbers  may  be  used  instead  of  those  in  the  text. 


'    HYDROSTATICS.  283 

chain;  and  -^— =a.!33=whole  height,  and    1.133— 1^.133. 
y 

Ans. 

2.  The  specific  gravity  of  lead  being  11.35  ;  of  cork,  .24  ;  of 
fir,  .45  :  How  much  cork  must  be  added  to  60  Ibs.  of  lead,  that 
the  united  mass  may  weigh  as  much  as  an  equal  bulk  of  fir  ? 

Let  #=weight  of  the  cork  in  pounds. 
Then  60+£=weight  of  united  mass=weight  of  an  equal  bulk 

of  fir.      .'. =do.  of  an  equal  volume  of  water. 

And  +— =the  same,  and  therefore =•         . 

Hence,  x=65.8527  Ibs.    Ans. 

3.  A  cone,  whose  height  is  6  inches,  is  immersed  in  water, 
with  its  vertex  downward.     Its  specific  gravity  being  |th  that 
of  the  water,  what  part  of  the  axis  will  be  immersed  ? 

Let  #=part  of  the  axis  immersed  : 

(91fi\  "?       f\ 
±^j    =-=0ne  half  the  axis.  Ans. 

4.  A  sailor  had  a  10  gallon  cask  half  full  of  brandy,  and  not 
being  allowed  to  keep  liquors  on  board,  he  threw  it  overboard 
for  the  purpose  of  concealment,  attaching  to  it  a  mass  of  lead 
just  sufficient  to  keep  it  under  water :  Required  the  weight  of  the 
lead,  the  wood  of  the  cask  containing  216  cubic  inches,  the  spe- 
cific gravity  of  the  brandy  being  .886,  that  of  sea  water  1.030, 
that  of  the  lead  11.35,  and  that  of  the  wood  .8  ? 

The  whole  quantity  of  water  displaced=231  x  10+216=2526 
inches  ;  and  231  X5=1155=quantity  of  brandy. 
Then,  1728  :  1030::  2526  :  1505.66=wt.  of  water. 
1728:     886  ::  1155  :     592.2  =wt.  of  brandy. 
1728:     800::   216  :     100.     =wt.  of  wood. 
Then    1505.66-  (592.2+100  =  692.2)  =813.46  ^weight,   of   the 
water  above  that  of  the  brandy  and  cask,  which  is  the  buoyant 
power  to  be  counteracted  by  the  lead. 

Now,  how  much  lead  will  it  take  to  weigh  813.46  in  water? 

Let  x= wt.  of  lead  required  ;  then  ———7= weight  of   an   equal 

1  l.UXi 

bulk  of  water.     Therefore  x—  —813.46,  and  z=894.64  oz. 

or  nearly  56  Ibs.  Ans. 

QUESTIONS   ON   HYDROSTATICS. 

1.  In  a  hydrostatic  press,  (Fig.   170,)  the  height  of  the  small 
column  AB  on  which  the  power  acts  is  2  feet  above  the  bottom 

*  Th*  weight  of  lead  above  sea  water. 


284  NATURAL,  PHILOSOPHY. 

of  the  large  piston  CD  ;  the  diameter  of  the  cylinder  AB  is  1 
inch,  and  of  the  cylinder  CD  1  foot.  By  means  of  a  screw  turned 
by  a  lever,  a  man  can  exert  a  force  on  A  equal  to  500  pounds  : 
What  amount  of  pressure  can  he  apply  with  the  aid  of  this  press, 
combining  his  own  strength  with  the  pressure  of  the  column  of 
water  AB  ?*  Ans.  72098.17  Ibs. 

2.  A  junk  bottle,  whose  lateral  surface  contained  50  square 
inches,  was  let  down  into  the  sea  3000  feet :  What  pressure 
would  the  sides  of  the  bottle  sustain,  no  allowance  being  made 
for  the  increased  specific  gravity  of  the  sea  water  ? 

Ans.  65104.166  Ibs. 

3.  A  Greenland  whale  sometimes  has  a  surface  of  3600  squaro 
feet :  What  pressure  would  he  bear  at  the  depth  of  800  fathoms  ? 

Ans.  1080,000,000  Ibs.,  or  more  than  482,142  tons. 

4.  A  mill-dam,  running  perpendicularly  across  a  river,  slopes 
at  an  angle  of  25  degrees  with  the  horizon.     The  average  depth 
of  the  stream  is  12  feet,  and  its  breadth  500  yards :  Required  the 
amount  of  pressure  on  the  dam  ? 

Ans.  15971906  Ibs.,  or  7130+  tons. 

5.  A  mineral  weighs  960  grains  in  air,  and  739  grains  in  wa- 
ter:  What  is  its  specific  gravity  ?  Ans.  4.344. 

6.  What  are  the   respective  weights  of  two  equal  cubical 
masses  of  gold  and  cork,  each  measuring  2  feet  on  its  linear  edge  ? 

Ans.  The  gold  weighs  9625  Ibs.  =4.297  tons ;  the  cork  weighs 
120  Ibs. 

7.  On  one  of  the  peaks  of  the  Alps,  is  a  single  mass  of  granite 
rock  of  a  nearly  globular  shape,  which  is  estimated  by  measure 
to  contain  5949  cubic  feet.     The  whole  mass  is  so  nicely  bal- 
anced on  its  center  of  gravity,  that  a  single  man  may  give  it  a 
rocking  motion.     By  trial  made  upon  a  small  fragment,  its  speci- 
fic gravity  was  found  to  be  2.6  :  What  is  its  weight  ? 

Ans.  431.568  tons. 

8.  An  island  of  ice  rises  30  feet  out  of  water,  and  its  upper 
surface  is  a  circular  plane,  containing  |ths  of  a  square  acre. 
On  the  supposition  that  the  mass  is  cylindrical,  required  its 
weight,  and  depth  below  the  water,  the  specific  gravity  of  sea 
water  being  1.0263,  and  that  of  ice  .92? 

Weight  242900  tons,-  depth  259.64 feet. 

9.  Wishing  to  ascertain  the  exact  number  of  cubic  inches  in  a 
very  irregular  fragment  of  stone,  I  ascertained  its  loss  of  weight 
in  water  to  be  5.346  ounces  :  Required  its  dimensions  ? 

Ans.  9.238  cubic  inches 

10.  Hiero,  king  of  Syracuse,  ordered  his  jeweller  to  make  him 
a  crown  of  gold  containing  63  ounces.     The  artist  attempted  a 
fraud  by  substituting  a  certain  portion  of  silver ;  which  being 

*  It  is  obvious  that  the  elevation,  and  consequently  the  pressure,  of  this  column 
would  be  continually  diminishing.  The  question  respects  only  the  commencement  of 
the  process. 


HYDRAULICS.  285 

suspected,  the  king  appointed  Archimedes  to  examine  it.  Archi- 
medes, putting  it  into  water,  found  it  raised  the  fluid  8.2245 
inches ;  and  having  found  that  the  inch  of  gold  weighs  10.36 
ounces,  and  that  of  silver  5.85  ounces,  he  discovered  what  part 
of  the  king's  gold  had  been  purloined :  It  is  required  to  repeat 
the  process  1  Ans.  28.8  ounces. 


CHAPTER  II. 

OF  LIQUIDS  OR  NON-ELASTIC  FLUIDS  IN  MOTION. 

415.  THAT  branch  of  Natural  Philosophy  which  treats  of  fluids 
in  motion,  is  usually  denominated  Hydraulics,  from  \>8ug,  water, 
and  auXocr,  a  torrent.     It  embraces,  therefore,  the  phenomena  ex- 
hibited by  water  issuing  from  orifices  in  reservoirs — projected 
obliquely  or  perpendicularly — flowing  in  pipes,  canals  and  rivers 
— oscillating  in  waves — or  opposing  a  resistan.ee  to  the  progress 
of  solid  bodies.* 

In  this  part  of  the  doctrine  of  fluids,  the  deductions  of  theory 
alone  are  of  little  value,  and  are  in  fact  so  discordant  with  expe- 
rience, that  little  reliance  can  be  placed  on  them,  except  as  modi- 
fied by  experiment.  When  thus  modified,  the  truths  learned 
respecting  the  laws  that  govern  the  motions  of  fluids,  have  a  high 
degree  of  practical  utility. 

The  causes  of  this  discordance  between  theory  and  experi- 
ment are  such  as  the  following :  the  different  states  in  which  the 
same  fluid  is  found  in  consequence  of  changes  of  temperature, 
and  different  degrees  of  purity — the  attraction  existing  between 
its  particles  and  preventing  that  perfect  fluidity  which  is  con- 
templated by  the  definition,  and  upon  which  our  reasonings  are 
founded — the  friction  against  the  sides  of  the  vessel — the  resist- 
ance of  the  air — the  size  of  the  vessel  in  proportion  to  the  aper- 
ture— the  shape  of  the  aperture  itself — the  different  directions 
in  which  the  various  parts  or  filaments  (as  they  are  called)  of  the 
fluid  run  toward  the  aperture — and  the  vortices,  or  irregular 
motions  which  are  communicated  to  the  fluid  by  a  variety  of 
causes,f 

416.  The  manner  in  which  vessels  of  water  discharge  them- 
selves through  small  orifices  in  the  bottom  or  sides  of  the  vessels, 
has  been  carefully  observed  by  introducing  into  a  column  of  wa- 
ter contained  in  a  glass  vessel,  small  solid  particles,  which  ren- 
ier  the  currents  of  the  fluid  visible.     From  such  observations  it 

*  Ed.  Encyc.,  Art.  Hydrodynamics.  t  Cavallo,  I,  267. 


280  NATURAL    PHILOSOPHY. 

appears,  that  the  particles  of  fluid  descend  in  vertical  lines,  until 
they  arrive  within  three  or  four  inches  of  the  orifice,  when  they 
gradually  turn  into  a  direction  more  or  less  oblique,  and  run  di- 
rectly toward  the  orifice.  When  the  surface  of  the  descending 
column  comes  very  near  to  the  orifice,  a  funnel-shaped  hollow  or 
cavity  makes  its  appearance,  and  the  various  particles  which 
rush  toward  the  orifice,  converge  to  a  point  without  the  orifice, 
at  the  distance  of  the  semi-diameter  of  the  orifice  itself,  where  is 
the  point  of  greatest  contraction,  called  the  vena  contracta. 

From  the  great  number  of  propositions  which  contain  the  doc- 
trine of  hydraulics,  we  shall  select  those  which  appear  to  be 
most  valuable  on  account  of  their  practical  utility,  adding  such 
remarks  as  will  serve  to  show  the  modifications  to  which  they 
are  subject  in  practice. 

417.  If  a  fluid  runs  through  any  tube,  pipe,  or  canal,  and  keeps 
it  constantly  full,  its  velocity,  in  any  part  of  its  course,  will  be  IN- 
VERSELY AS  THE  AREA  OP  THE  SECTION  at  that  part. 

Let  A  and  a  be  the  areas  of  two  cross  sections  of  a  tube  of 
unequal  bore,  and  let  V  and  v  denote  the  velocities  of  the  fluid 
as  it  flows  through  A  and  a  respectively.  Now  the  quantity  of 
fluid  which  passes  through  any  section,  must  depend  upon  the 
area  and  velocity  conjointly  ;  and  since  the  tube  is,  by  the  sup- 
position, kept  constantly  full,  the  same  quantity  runs  through 
every  section  of  the  tube  in  any  given  time.  Hence  AxV=axu 
.*. A :  a  : :  v  :  V  ;  that  is,  the  velocity  is  inversely  as  the  area  of  the 
section. 

It  follows  from  the  foregoing  proposition,  that  the  velocity  of  a 
stream  increases  as  either  the  breadth  or  the  depth  decreases. 
In  a  tube,  the  parts  near  the  axis  move  with  greater  velocity 
than  near  the  sides ;  and  in  every  canal  or  river,  the  velocity  of 
the  stream  is  greater  in  the  central  parts,  or  channel,  than  at  the 
sides,  and  greater  at  the  surface  than  at  the  bottom.  The  mean 
velocity,  therefore,  is  to  be  inferred  from  the  combination  of  at 
least  three  distinct  measurements.  For  example,  the  velocity  oi 
a  stream  was  found  to  be 

On  the  surface  of  the  channel,     -    -     5  miles  per  hour. 

At  the  bottom, -     3      "          " 

Af  the  sides, 3£     "          " 

Therefore,  the  mean  velocity = —=—^-=3.83  miles   per 

o  o 

hour. 

It  is  important  to  avoid  all  unnecessary  expansions  as  well  as 
contractions,  in  pipes  or  canals,  since  there  is  always  a  useless 
expenditure  of  force  in  restoring  the  velocity  which  is  lost  in  the 
wider  parts.* 

*  Young's  Lectures  on  Nat.  Phil.,  I,  283. 


HYDRAULICS.  287 

418.  To  find  the  quantity  of  water  which  flows  in  a  river,  we 
may  first  ascertain  the  area  of  a  section  by  taking  the  depth  in 
various  parts  of  the  section,  and  dividing  the  sum  of  all  the 
depths  by  the  number  of  measurements,  which  gives  us  the  mean 
depth.     This  multiplied  into  the  breadth  of  the  stream,  gives  the 
area  of  the  section,  which  multiplied  into  the  average  velocity, 
(Art.  417,)  shows  the  quantity  required. 

Example. — Wishing  to  know  the  quantity  of  water  that  dis- 
charged itself  per  minute  through  a  certain  strait,  I  chose  a 
place  below  the  strait  where  the  water  flowed  at  a  nearly  uni- 
form rate,  where,  by  noting  the  time  occupied  by  light  sub- 
stances in  floating  through  various  parts  of  the  stream  for  a  given 
distance,  I  found  the  average  velocity  to  be  96  feet  per  minute. 
The  mean  depth  of  a  section  was  8^  feet,  and  the  breadth  560 
feet.  Hence, 

8^x560x96—456960  cubic  feet  per  minute. 

419.  The  phenomena  of  RIVERS  have  sometimes  been  explain- 
ed on  the  supposition  that  rivers  are  bodies  falling  freely  down 
inclined  planes.     But  the  conclusions  deduced  from  this  doctrine 
are  so  at  variance  with  experience,  as  to  be  of  no  value.*   Were 
every  part  of  the  bed  of  a  river  uniform,  like  a  tube,  the  chan- 
nel or  portion  which  moves  with  the  greatest  velocity,  would  be 
in  the  center  of  the  surface  ;  but  inequalities  in  the  sides  and 
bottom  usually  throw  it  out  of  the  center,  and  incline  it  to  one 
side  or  the  other.     The  increased  velocity  of  a  stream  during  a 
freshet,  while  the  stream  is  confined  within  its  banks,  exhibits 
something  of  the  acceleration  which  belongs  to  bodies  falling 
freely  down  an  inclined  plane.     It  presents  the  case  of  a  river 
flowing  upon  the  top  of  another  river,  and  consequently  meeting 
with  much  less  resistance  than  when  it  runs  upon  the  rough  un- 
equal surface  of  the  earth  itself.     The  augmented  force  of  a 
stream  in  a  freshet,  arises  from  the  simultaneous  increase  of  the 
quantity  of  water  and  the  velocity.     In  consequence  of  the  fric- 
tion of  the  banks  and  beds  of  rivers,  and  the  numerous  obstacles 
they  meet  with  in  their  winding  course,  their  progress  is  very 
slow,  whereas,  were  it  not  for  these  impediments,  it  would  be- 
come immensely  great,  and  its  effects  would  be  exceedingly  dis- 
astrous.    A  very  slight  declivity  is  sufficient  for  giving  the  run- 
ning motion  to  water.     Three  inches  per  mile,  in   a   smooth, 
straight  channel,  gives  a  velocity  of  about  three  miles  per  hour. 
The  Ganges,  which  gathers  the  waters  of  the  Himalaya  Moun- 
tains, the  loftiest  in  the  world,  at  the  distance  of  eighteen  hun- 
dred miles  from  its  mouth,  is  only  eight  hundred. feet  above  the 
level  of  the  sea, — that  is,  about  twice  the  height  of  St.  Paul's 
church  in  London  ;   and  to  fall  these  eight  hundred  feet,  in  its 

*  See  Robison's  Mech.  Phil.,  a:  Encyc.  Brit.,  Art.  Theory  of  Rivera. 


288  NATURAL   PHILOSOPHY. 

long  course,  the  water  requires  more  than  a  month.  The  great 
river  Magdalena,  in  South  America,  running  for  a  thousand  miles 
between  two  ridges  of  the  Andes,  falls  only  five  hundred  feet  in 
all  that  distance.*  The  Croton  aqueduct,  that  waters  the  city 
of  New  York,  falls  one  foot  per  mile.  The  motion  of  rivers  soon 
becomes  uniform,  even  in  very  great  inclinations,  the  sum  of  the 
resistances  forming  an  equilibrium  with  the  acceleration  pro- 
duced by  the  descent  on  an  inclined  plane.  The  smallest  incli- 
nation capable  of  giving  motion  to  water  is  a  descent  of  one 
foot  in  a  million  ;  or  about  ^jth  of  an  inch  per  mile.  A  fall  of  3 
feet  per  mile,  makes  a  mountain  torrent,  f 

420.  The  velocity  with  which  a  fluid  issues  from  a  small  orifice 
in  the  bottom  or  side  of  a  vessel,  kept  constantly  full,  varies  as  th" 
SQUARE  ROOT  OF  THE  DEPTH  below  the  surface  of  the  fluid. 

The  pressure  at  different  depths  varies  as  the  depth,  (Art.  385 ;) 
also  the  momentum  is  as  the  force  impressed,  and  likewise  there- 
fore as  the  depth.  Hence,  let  Q,  Q',  denote  the  quantities  of  fluid 
discharged  at  each  orifice  respectively ;  V,  V,  the  corresponding 
velocities,  and  AB,  AC,  the  depths.  Then,  since  the  momentum 
isasQxV,  QxVrQ'xV'::  AB:  AC. 

But,  through  a  given  orifice,  the  quantity  of  fluid  discharged 
evidently  varies  as  the  velocity,  or  Q,  <x  V ;  hence, 
V2:V'2::    AB:      AC,  or 
V  :  V  ::v/AB:  VAC. 

It  appears,  therefore,  that  the  fluid  issues  with  the  velocity 
which  a  body  would  acquire  by  falling  freely  from  the  surface  of 
the  fluid  to  the  orifice,  for  that  is  also  as  the  square  root  of  the 
space.  (Art.  29.)  And  since  a  body  when  projected  upward 
with  a  certain  velocity,  will  rise  to  the  same  height  as  that  from 
which  it  must  have  fallen  to  acquire  that  velocity,  consequently, 
if  a  fluid  issue  from  the  side  of  a  vessel  through  a  spout  bent  up- 
ward, it  would,  were  it  not  for  the  resistance  of  the  air,  and  fric- 
tion at  the  orifice,  rise  to  the  level  of  the  fluid  in  the  reservoir. 

It  follows  from  the  foregoing  proposition,  that  an  orifice  six- 
teen inches  from  the  surface  will  discharge  twice  as  much  fluid 
in  a  given  time  as  one  at  the  depth  of  four  inches  ;  and  this  is 
conformable  to  the  law  that  the  pressure  varies  as  the  depth ;  for 
since  twice  the  quantity  flows  with  twice  the  velocity,  of  course 
the  pressure  or  momentum  is  four  times  as  great  at  the  depth  of 
sixteen  as  at  that  of  four  inches.  In  order,  therefore,  to  draw 
off  from  a  cistern  four  times  as  much  water  as  before,  we  must 
place  the  gate  sixteen  times  as  deep.  A  gate  opened  in  a  reser- 
voir at  the  depth  of  sixty  four  inches,  will  discharge  only  four 
times  as  much  as  at  the  depth  of  four  inches. 

421.  In  the  construction  of  water  works  it  is  customary  to  con- 
*  Arnott's  El.  Phys.,  I,  260.  t  Rennie,  Reports  of  Brit.  Assoc.,  1834. 


HYDRAULICS.  289 

duct  the  stream,  or  such,  part  of  it  as  is  required,  into  a  cubical 
cistern,  and  to  let  it  issue  from  the  side  of  this,  near  to  the  bot- 
tom, and  thus  fall  upon  the  main  wheel.  Instead  of  admitting 
the  water  to  the  wheel  in  this  manner,  it  has  sometimes  been 
supposed  that  an  advantage  might  be  gained  by  letting  the  water 
fall  down  a  height  equal  to  that  of  the  top  of  the  cistern,  per- 
pendicularly upon  the  wheel,  on  the  supposition  that  we  might 
thus  avail  ourselves  of  the  force  acquired  by  the  water  in  falling. 
But  according  to  the  preceding  proposition,  the  force  would  be 
the  same  whether  the  water  issued  from  the  cistern  and  thus  ap- 
plied itself  to  the  wheel,  or  whether  it  fell  upon  the  wheel  from 
a  height  equal  to  that  of  the  surface  of  the  wrater  in  the  reser- 
voir above  the  orifice.  This  is  true  in  theory,  but  in  practice  it 
would  be  found  more  advantageous  to  take  the  water  out  of  the 
cistern,  to  avoid  loss  from  the  resistance  of  the  air. 

422.  If  a  cylindrical  or  prismatic  vessel,  of  which  the  horizontal 
section  is  everywhere  the  same,  is  filled  with  fluid,  and  empties  it- 
self by  an  orifice,  the  velocity  with  which  the  surface  descends,  and 
also  the  velocity  with  which  the  water  issues,  is  uniformly  retarded. 

The  velocity  with  which  the  surface  descends  is  proportional 
to  that  with  which  the  fluid  issues  from  the  orifice,  and  therefore 
is  as  the  square  root  of  the  depth.  (Art.  420.)  But  the  velocities 
of  bodies  projected  perpendicularly  upward  are  in  the  same  ratio 
to  their  spaces,  (Art.  30,)  and  therefore  a  body  projected  perpen- 
dicularly upward  is  in  the  same  relative  circumstances  as  the 
descending  surface  of  the  fluid ;  and  as  the  projected  body  is 
uniformly  retarded,  the  same  is  true  of  the  descending  surface. 

On  this  principle  is  constructed  the  Clepsydra,  or  water-clock. 
Since  the  descent  of  the  surface  is  uniformly  retarded,  the  spaces 
which  it  describes  in  equal  times,  reckoning  from  the  bottom,  are 
as  the  odd  numbers,  1,  3, 5,  7,  &c.  ;  and  if  a  cylindrical  vessel  of 
water  be  furnished  with  an  orifice  at  the  bottom  which  will  ex- 
actly discharge  the  whole  column  in  twelve  hours,  and  the  sides 
of  the  vessel  be  divided  into  spaces  corresponding  to  the  forego- 
ing numbers,  the  successive  heights  of  the  column  become  meas- 
ures of  time.* 

423.  If  we  accurately  mark  the  time  in  which  a  cylindrical  or 

£ismatic  vessel,  whose  horizontal  section  is  everywhere  the  same, 
charges  itself  to  the  level  of  a  given  orifice,  and  then  draw  off 
for  the  same  time,  keeping  the  vessel  constantly  full,  we  shall  obtain 
double  the  quantity  of  fluid  in  the  latter  case  as  in  the  former. 

When  the  vessel  is  kept  constantly  full,  the  velocity  at  the 
orifice  (and  of  course  the  quantity  discharged)  continues  uni- 

*  An  excellent  description  and  representation  of  various  forms  of  the  Clepsjdra 
may  be  found  in  the  Encyclopaedia  Metropolitana,  Art.  Hydrodynamics 

37 


290  NATURAL   PHILOSOPHY. 

formly  the  same  as  at  first  ;  and  since  the  circumstances  of  this 
case  are  exactly  analogous  to  those  of  a  body  projected  perpen- 
dicularly upward  ;  and  since  if  a  body  thus  projected  were  to 
continue  to  ascend  with  the  first  velocity,  it  would  pass  over  a 
space  twice  as  great  in  the  same  time  as  when  uniformly  re- 
tarded ;  therefore,  the  truth  of  the  proposition  is  manifest. 

424.  A  fluid  spouting  from  the  side  of  a  vessel,  describes  the 
curve  of  a  parabola. 

The  fluid  is  precisely  in  the  same  circumstances  as  a  projec- 
tile acted  on  by  the  force  of  projection  (viz.  the  pressure  of  the 
incumbent  fluid)  and  by  the  force  of  gravity.  Therefore,  ac- 
cording to  Art.  48,  it  describes  the  curve  of  a  parabola.  As  in 
the  case  of  other  projectiles,  the  proposition  holds  good,  whatever 
may  be  the  angle  of  elevation  of  the  jet. 

425.  If  a  semicircle  be  described  on  the  perpendicular  side  of  a 
vessel,  as  a  diameter,  and  a  fluid  spout  horizontally  from  any  point 
in  the  diameter,  its  random  will  equal  twice  the  length  of  the  ordi- 
nate  to  that  point. 

The  velocity  with  which  the  fluid 
issues  from  the  vessel,  being  that 
which  is  due  to  the  height  BG, 
(Fig.  186,)  is  such  as  if  preserved 
\vould  carry  the  jet  through  a  space 
equal  to  2BG,  in  the  time  of  the  fall 
through  BG;  but  after  leaving  the 
orifice,  it  arrives  at  the  horizontal 

plane  in  the  same  time  as  that  in  __ 
which    a  body    would    fall    freely  c 

through  GD.  And  since,  in  falling  bodies,  the  times  are  as  the 
square  roots  of  the  spaces,  (Art.  29,)  therefore,  VBG  :  v/GD  :  : 
tBG  :  tGD*  that  is,  the  time  employed  in  reaching  the  plane.f 
But  in  the  time  of  describing  BG,  the  jet  would  be  carried  uni- 
formly and  horizontally  over  a  space  equal  to  2BG  ;  therefore, 
to  find  how  far  it  will  proceed  in  a  horizontal  direction,  while 
it  is  descending  through  GD,  we  have 


v/BG  :  VGD  ::  2BG  :  DE-  =2V  (BG  xGD)=2GH, 


that  is,  double  the  ordinate  to  the  point  G. 

The  greatest  random  will  be  when  the  fluid  spouts  from  the 
center,  for  then  the  ordinate  (and  of  course  twice  the  ordinate)  is 


*  t  signifies  the  time  of  falling  down  the  spaces  BG  and  GD. 

t  For  the  jet  will  reach  the  plane  in  the  curve,  acted  on  by  the  two  forces  of  pro- 
jection and  gravity,  in  the  same  time  in  which  it  would  descend  through  the  perpen 
dicular  height,  urged  by  gravity  alone. 


HYDRAULICS. 


291 


greatest ;  and  the  randoms  will  be  equal  at  equal  distances  above 
and  below  the  center,  for  at  these  points  the  ordinates  are  equal. 

426.  When  a  fluid  is  contained  in  any  vessel,  it  presses  equally 
on  opposite  points  of  the  vessel,  and  is  thus  maintained  at  rest. 
Now  if  we  remove  the  pressure  from  one  of  the  opposite  points 
while  it  remains  on  the  other,  the  force  exerted  on  the  latter 
will  tend  to  move  the  vessel  in  that  direction.  Thus  if  we  sus- 
pend a  bottle  of  water,  like  a  pendulum,  and  open  a  small  jet  on 
one  side,  the  pressure  on  this  side  being  taken  off,  but  still  re- 
maining on  the  other,  the  bottle  M  ill  swing  toward  the  side  op- 
posite the  orifice,  and  remain  suspended  at  a  certain  height 
above  its  former  position. 

Barker's    Mill    acts  on   the  p;g   !87 

foregoing  principle.*     Its  con-  -^sus— — *  *  i  _    i  , 

.struction  is  as  follows  :  AB 
(Fig.  187)  is  a  hollow  cylinder 
movable  about  a  vertical  axis 
MN.  PP'  is  another  cylinder 
placed  at  right  angles  to  the 
former,  and  communicating  in- 
ternally \vith  it.  Near  its  ex- 
tremities, which  are  closed,  two 
apertures  are  made  in  the  sides 
of  this  horizontal  cylinder,  open- 
ing in  opposite  directions.  That 
at  P  is  supposed  to  front  the 
reader,  and  that  at  P'  to  lie  on" 
the  opposite  side  of  the  tube. 
Water  is  supplied  by  the  spout  O,  keeping  AB  constantly  nearly 
full.  As  the  water  flows  out  at  P  and  P',  the  unbalanced  pres- 
sure on  the  sides  of  the  cylinder  opposite  to  those  openings,  acts 
on  the  respective  arms  PB,  P'B,  and  sets  the  horizontal  cylinder 
revolving,  carrying  along  with  it  the  perpendicular  shaft  MN, 
and  any  machinery  connected  with  it.  The  power  resulting 
from  the  pressure  of  a  column  of  water  is  here  applied  to  the 
best  advantage,  for  since  the  arms  of  the  horizontal  shaft  BP 
and  BP'  may  be  lengthened  at  pleasure,  while  the  power  is  still 
applied  at  the  extremity  of  each,  the  circumstances  are  the  same 
as  when  the  power  is  applied  to  the  end  of  a  lever  or  the  cir- 
cumference of  a  wheel,  and  the  power  gains  a  similar  advantage. 
Moreover,  the  centrifugal  force  acquired  by  the  revolving  fluid, 
being  greatest  at  the  extremities  of  the  shaft,  acts  under  a  simi- 
lar advantage  and  conspires  with  the  simple  pressure.  This 
machine  is  said  by  writers  on  mechanics  to  be  the  most  effective 


*  Machines  acting  on  tills  principle,  are  erroneously  said  to  go  by  reaction. 


292  NATURAL   FHILOSOPHY. 

known  for  applying  the  power  of  a  given  quantity,  and  a  given 
fall  of  water,  to  the  working  of  machinery.* 

427.  The  term  FRICTION  is  applied  to  the  obstruction  occa- 
sioned to  the  passage  of  fluids  in  the  same  manner  as  it  is  to  sol- 
ids ;-  and  it  exists  to  such  an  extent  as  to  become  an  object  of 
considerable  inconvenience  in  practice.     It  can  be  obviated  only 
by  making  the  conveying  pipe  of  much  larger  dimensions  than 
would  otherwise  be  necessary,  so  as  to  allow  the  free  passage  of 
a  sufficient  quantity  of  fluid  through  the  center  of  the  pipe,  while 
a  ring  or  hollow  cylinder  of  water  is  considered  to  be  at  rest  all 
around  it.     Other  circumstances  beside  friction  likewise  tend  to 
diminish  the  quantity  of  fluid  which  would  otherwise  pass  through 
pipes, — such  as  the  existence  of  sharp  or  right-angled  turns  in 
them,  permitting  eddies  or  currents  to  be  formed,  or  not  provid- 
ing for  the  eddies  or  currents  that  form  naturally,  by  suiting  the 
shape  of  the  pipe  to  them.     It  follows,  therefore,  that  whenever 
a  bend  or  turn  is  necessary  in  a  water-pipe,  it  should  be  made  in 
as  gradual  a  curve  or  sweep  as  possible  ;  that  the  pipe  should 
not  only  be  sufficiently  capacious  to  afford  the  necessary  supply 
but  should  be  of  a  uniform  bore  throughout,  and  free  from  all 
projections  or    irregularities  against  which  water  can  strike, 
and  form  eddies  or  reverberations,  since  these  will  impede  the 
progress  of  the  fluid  as  effectually  as  the  most  solid  obstacles. 

428.  Fluids,  it  must  be  recollected,  are  subject  to  the  same 
law  of  gravity  as  solid  bodies,  and  a  mass  of  fluid  descending 
vertically  has  its  motion  accelerated  in  the  same  manner  as  a 
solid  mass  ;  and  the  momentum  generated  is  the  product  of  its 
quantity  of  matter  and  velocity.     If  a  column  of  water  move 
through  either  a  vertical  or  an  inclined  pipe,  it  acquires  a  velo- 
city, which  from  the  friction  of  the  pipe  will  soon  become  uni- 
form, and  the  momentum  generated  will  be  measured  by  the 
mass  multiplied  into  this  uniform  velocity.     Now  force  is  also 
necessary  for  the  destruction  of  motion,  and  the  shorter  the  time 
through  which  it  acts  the  greater  is  the  effect  produced.     Thus 
a  small  hammer  with  a  hard  face  is  much  more  effective  in 
driving  a  nail,  than  a  mallet  of  twenty  times  its  weight,  and 
moved  with  the  same  velocity.     For  in  consequence  of  the  hard- 
ness of  the  face,  the  motion  is  destroyed  instantly  and  is  instantly 
received  by  the  nail.     By  this  means  the  momentum  is  confined 
to  the  nail,  whereas  were  the  motion  communicated  gradually, 
it  would  be  diffused  more  or  less  over  the  body  into  which  the 
nail  is  driven.  (See  Arts.  231,  250.)     The  sudden  destruction  of 
motion  in  a  fluid  mass  is  attended  with  effects  precisely  analo- 
gous.    When  the  motion  of  a  large  mass  of  water  is  suddenly 

*  Mosely's  Mechanics  applied  to  the  Arts,  p.  231. 


HYDRAULICS.  293 

stopped,  the  surface  which  stops  it  sustains  a  very  great  force. 
The  operation  of  this  principle  is  seen  when  the  gates  of  a  lock 
are  instantly  closed,  and  when  the  stop-cock  of  an  aqueduct 
which  discharges  a  rapid  jet  of  water,  is  suddenly  shut.  In  the 
latter  case,  the  violence  is  sometimes  such  as  to  burst  the  pipe 
nearest  the  opening.  A  powerful  engine  for  raising  water, 
called  the  Hydraulic  Ram,  acts  on  this  principle.* 

429.  An  unexpected  facility  is  gained  in  the  discharge  of  a 
fluid  from  the  bottom  or  side  of  a  vessel,  by  applying  a  pipe  to 
the  orifice.     On  account  of  the  friction  known  to  occur  in  the 
passage  of  a  fluid  through  a  tube,  it  might  be  supposed  that  a 
simple  orifice  made  in  the  vessel  might  be  more  favorable  to  the 
discharge  of  the  fluid  than  an  opening  prolonged  by  a  tube  ;  but 
it  has  been  found  by  experiment,  that  a  vessel  of  tin,  with  a 
smooth  hole  formed  in  its  bottom,  did  not  discharge  water  as 
rapidly  as  another  containing  the  same  weight  of  water,  and  an 
orifice  of  the  same  dimensions,  to  which  a  short  pipe  was  applied. 
By  varying  the  length  of  the  pipe,  it  is  found  that  when  its  length 
is  twice  its  diameter,  it  produces  the  most  rapid  discharge,  deliv- 
ering in  this  case  82  quarts  of  water  in  100  seconds,  while  the 
simple  hole  delivered  but  62  quarts  in  the  same  time.     If,  how- 
ever, the  pipe  projects  into  the  vessel,  the  quantity  discharged  is 
diminished  instead  of  being  increased  by  the  pipe.f 

When  water  is  conveyed  through  a  straight  cylindrical  pipe 
of  any  length,  the  discharge  of  water  may  be  increased  by  only 
altering  the  shape  of  the  terminations  of  that  pipe,  viz.  by  making 
the  end  of  the  pipe  which  is  close  to  the  reservoir,  or  the  entrance 
to  it,  of  the  conical  shape  of  the  vena  contracta,  (Art.  416,)  and 
by  making  the  other  extremity  of  the  pipe,  where  the  water  is- 
sues, of  a  trumpet-shape.J  By  this  means,  the  quantity  of  water 
which  is  discharged  in  a  given  time,  is  more  than  doubled.§ 

WATER   WHEELS. 

430.  Three  kinds  of  water  wheels  are  employed  under  differ- 
ent circumstances,  namely,  the  overshot,  the  undershot,  and  the 
breast  wheel. 

The  overshot  wheel  is  used  when  the  supply  of  water  is  scan- 
ty, since  this  construction  admits  of  a  more  economical  use  of  the 

*  See  Webster's  Principles  of  Hydrostatics,  p.  145. 

t  Ed.  Encyc.,  Art.  Hydrodynamics.— Millington's  Epitome  of  Nat.  Phil.,  181. 

\  Experience  shows  that  the  divergency  of  this  termination  must  not  exceed  a  cer- 
tain degree,  for  in  that  case  it  will  prove  rather  disadvantageous  than  useful.  It  ap- 
pears that  when  the  divergency  is  greater  than  an  angle  of  16  degrees,  the  effect 
ceases  entirely ;  and  that  the  greatest  quantity  of  fluid  is  discharged,  when  the  di- 
vergency is  equal  to  an  angle  of  about  3  degrees. — Cavallo. 

§  Cavallo,  I,  276.  See  Ewbank's  Hydraulics.— Rennie's  Report  to  the  British 
Association,  1834. 


294 


NATURAL   PHILOSOPHY. 


water  than  either  of  the  others.  Fig.  188,  represents  a  section  of 
an  overshot  wheel  at  right  angles  to  the  axis.  Its  diameter  is 
usually  nearly  equal  to  the  whole  fall  of  the  water.  It  is  placed 
under  the  head  of  water  in  such  a  way  as  to  receive  its  whole 
force  into  buckets,  connected  with  the  rim  of  the  wheel.  These 
buckets  are  made  of  such  a  shape  as  to  retain  as  much  of  the 
water  as  possible  until  they  reach  Fig  188 

the  lowest  point  of  the  wheel, 
but  none  at  all  after  passing  that 
point.  43y  this  means  the  weight 
of  the  water  in  the  buckets  is 
made  to  exert  as  much  weight 
as  possible  on  one  side  of  the 
wheel,  thus  causing  it  to  descend, 
while  they  oppose  little  resistance 
to  the  ascent  of  the  opposite  side 
of  the  wheel.  Let  us  trace  the 
effect  of  a  single  bucket  in  its 
revolution.  Were  it  to  receive 
the  water  directly  on  the  top  at 
H,  the  only  effect  would  be  to 
cause  an  increased  pressure  on  the  axis  of  the  wheel,  while  it 
would  not  contribute  to  turn  the  wheel ;  and,  indeed,  within  a 
certain  distance  from  H  towards  a,  the  weight  of  the  water  in- 
creases the  resistance  from  friction  on  the  axle  more  than  its 
force  tends  to  turn  the  wheel.  It  is  evident,  therefore,  that  the 
water  must  begin  to  fall  on  the  wheel  so  far  toward  the  side, 
that  its  leverage,  measured  from  O  on  the  line  OF,  may  enable 
it  to  overcome  the  friction  and  all  other  impediments.  As  the 
wheels  revolve,  the  weight  of  the  water  acts  with  a  constantly 
increasing  leverage,  until  at  F  it  acts  with  its  greatest  force. 
From  this  point,  the  force  declines  from  two  causes,  namely,  the 
loss  of  water  from  the  buckets  as  their  position  is  gradually  re- 
versed, and  the  diminution  of  the  leverage  or  effective  distance 
from  O  on  the  line  OF,  until,  before  it  reaches  the  lowest  point 
L,  it  may  again  slightly  act  as  an  impediment  by  increasing, 
from  its  weight,  the  friction  on  the  axle  more  than  it  contributes 
to  turning  the  wheel. 

431.  There  is  a  certain  velocity  with  which  an  overshot  wheel 
should  move  in  order  to  produce  the  greatest  effect.  If,  on  the 
one  hand,  the  wheel  is  loaded  so  heavily  that  the  weight  of  water 
is  insufficient  to  move  it,  then  of  course  the  effect  is  nothing ; 
and  if,  on  the  other  hand,  the  velocity  of  the  wheel  were  to  equal 
that  with  which  water  would  fall  freely,  then  its  pressure  on  the 
buckets  would  become  nothing  and  its  moving  power  nothing. 
The  best  velocity  that  can  be  given  to  an  overshot  wheel  is  found 
to  be  three  feet  per  second. 


HYDRAULICS. 


295 


432.  The  undershot  wheel  (Fig.  Fig.  189. 

x89)  is  carried,  not  by  the  weight 
of  the  water  simply,  as  is  the 
case  in  the  overshot  wheel,)  but 
by  the  momentum  or  force  of 
running  water.  Instead  of  close 
buckets  for  holding  wrater,  it  is 
furnished  with  float  boards,  which 
receive  the  impulse  of  the  stream. 
When  these  are  placed,  as  in  the  figure,  with  their  planes  at 
right  angles  to  the  rim  of  the  wheel,  the  latter  may  turn  either 
way ;  and  this,  therefore,  is  the  form  of  wheels  employed  in  tide 
mills.  When  the  wheel  is  required  to  turn  only  in  one  direction, 
an  advantage  is  gained  by  placing  the  float  boards  so  as  to  pre- 
sent an  acute  angle  toward  the  current,  by  which  means  the 
water  acts  partly  by  its  weight,  as  in  the  overshot  wheel.  The 
undershot  wheel  is  adapted  to  situations  where  the  supply  of 
water  is  always  abundant.  It  acts,  moreover,  with  the  greatest 
effect,  when  its  velocity  is  half  that  of  the  stream. 

Fig.  190. 


433.  The  breast  wheel  (Fig.  190)  combines  the  advantages  of 
both  the  others,  and  is  therefore  adapted  to  situations  where  the 
supply  of  water  is  generally  sufficient,  but  not  always  abundant. 
The  planes  of  the  float  boards  are  at  right  angles  to  the  rim  of 
the  wheel,  and  are  brought  so  near  to  the  mill  course  that  the 
float  boards  hold  water  like  buckets. 

According  to  Smeaton,  the  effect  of  overshot  wheels,  under  the 
same  circumstances  of  quantity  and  fall,  is,  at  a  medium,  double 
that  of  an  undershot  wheel. 


296  NATURAL  PHILOSOPHY. 


CHAPTER  III 

OF  CAPILLARY  ATTRACTION,  RESISTANCE  OF  FLUIDS,  AND 
WAVES 

434.  THE  definition  of  a  fluid,  (Art.  376,)  proceeds  on  the  sup- 
position that  fluids  are  destitute  of  cohesion,  and  that  their  par- 
ticles move  among  themselves  without  the  slightest  impediment. 
All  liquids,  however,  have  in  fact  more  or  less  cohesion  or  mutual 
attraction  among  their  particles.     This  is  apparent  in  their  form- 
ing drops,  and  in  the  viscidity  of  certain  liquids,  as  oil  and  tar, 
which  on  account  of  this  property  are  sometimes  denominated 
semi-fluids.     It  is  owing  to  this  property  that  water  so  readily 
forms  itself  into  drops,  and  that  its  surface  when  viewed  in  a 
small  cup  or  wine  glass,  appears  convex.     Both  of  these  proper- 
ties are  still  more  observable  in  quicksilver,  which  when  poured 
on  a  table,  forms  numerous  globules  of  a  perfectly  spherical 
figure  ;  and  the  convex  figure  of  the  surface,  as  seen  in  a  wine 
glass,  is  very  striking.     When  we  dip  a  glass  tube  into  water,  it 
comes  out  covered  with  drops  of  the  fluid,  which  are  held  by  the 
attraction  of  the  glass  for  water  ;  but  the  tube  when  dipped  into 
quicksilver  comes  out  dry,  because  the  cohesion  between  the 
particles  of  quicksilver  for  one  another  is  greater,  than  the  mu- 
tual attraction  that  exists  between  the  metal  and  the  glass. 
Hence,  a  solid  body,  when  immersed  in  a  fluid,  is  sometimes  wet 
by  it  and  sometimes  not,  according  as  the  attraction  between  the 
solid  and  the  fluid  is  greater  or  less  than  that  which  exists  be- 
tween the  particles  of  the  fluid  for  one  another. 

435.  If  a  disk  or  thin  plate  of  almost  any  solid  substance,  as 
of  glass  or  metal,  be  suspended  from  the  arm  of  a  balance  and 
counterpoised,  upon  bringing  it  into  contact  with  the  surface  of 
a  fluid  capable  of  wetting  it,  it  will  adhere  with  a  considerable 
force,  the  amount  of  which  will  be  indicated  by  the  weights  re- 
quired to  be  added  to  the  opposite  scale,  in  order  to  detach  it. 
This  experiment  shows  that  a  lamina  of  water  is  held  to  a  con- 
tiguous lamina  by  a  strong  force,  and  hence  when  a  cause  ope- 
rates upon  the  surface  of  a  fluid  to  draw  up  the  lamina,  a  column 
of  fluid  may  rise  along  with  it,  in  consequence  of  the  mutual 
cohesion  of  the  successive  laminae.     We  see  the  same  principle 
strikingly  exemplified  in  viscid  fluids,  as  tar,  where,  on  drawing 
up  a  small  portion  of  the  surface,  a  column  of  the  fluid  follows 
it.     The  foregoing  fact  will  lead  us  to  an  understanding  of  the 
causes  of  capillary  attraction. 


HYDRAULICS.  297 

436.  CAPILLARY  ATTRACTION  is   the  attraction  which  causes  the 
ascent  of  fluids  in  small  tubes. 

The  tubes  must  be  less  than  one  tenth  of  an  inch  in  diameter, 
and  tubes  whose  bores  are  no  larger  than  a  hair,  (capillus,)  present 
the  phenomenon  the  most  strikingly.  But  though  the  rise  of 
water  above  its  natural  level,  is  most  manifest  in  small  tubes,  it 
appears,  in  a  degree,  in  all  vessels  whatsoever,  by  a  ring  of  water 
formed  around  the  sides  with  a  concavity  upward.* 

The  following  are  the  leading  facts  respecting  capillary  at- 
traction. 

(1.)  When  small  tubes,  open  at  both  ends,  are  immersed  perpen- 
dicularly in  any  liquid,  the  liquid  rises  in  them  to  a  height  which  is 
inversely  as  the  diameter  of  the  bore.  Though  tubes  of  glass  are 
usually  employed  in  experiments  on  this  subject,  yet  tubes  made 
of  any  other  material  exhibit  the  same  property.  Nor  does  the 
thickness  of  the  solid  part  of  the  tube,  or  its  quantity  of  matter, 
make  the  least  difference,  the  effect  depending  solely  on  the  at- 
traction of  the  surface,  and  consequently  extending  only  to  a 
very  small  distance. 

(2.)  Different  fluids  are  raised  to  unequal  heights  by  the  same 
tube.  Thus,  according  to  Gay  Lussac,  a  tube  which  will  raise 
water  23  inches  will  raise  alcohol  only  9  inches. 

(3.)  A  tube  T  I-Q  of  an  inch  in  diameter  raises  water  5.3  inches , 
and  since  the  height  is  reciprocally  as  the  diameter,  the  product 
of  the  diameter  into  the  height  is  a  constant  quantity,  namely,  the 
.053th  part  of  an  inch  square,  f 

(4.)  Fluids  rise  in  a  similar  manner  between  plates  of  glass, 
metal,  fyc.,  placed  perpendicularly  in  the  fluids,  and  near  to  one 
another.  If  the  plates  are  parallel,  the  height  to  which  a  fluid 
will  rise,  is  inversely  as  the  distance  between  the  plates ;  and  the 
whole  ascent  is  just  half  that  which  takes  place  in  a  tube  of  the 
same  diameter.  If  the  plates  be  placed  edge  to  edge,  so  as  to  form 
an  angle,  and  they  be  immersed  in  water,  with  the  line  of  their 
intersection  vertical,  the  water  will  ascend  between  them  in  a 
curve  having  its  vertex  at  the  angle  of  intersection.  This  curve 
is  found  to  have  the  properties  of  the  hyperbola. 

(5.)  If  a  capillary  glass  tube  be  immersed  in  mercury,  the  mer- 
cury, instead  of  rising,  sinks  to  a  lower  level  within  than  without, 
and  its  surface  is  convex  instead  of  concave. 

(6.)  Tubes  which  are  wider  at  bottom  than  at  top,  elevate  fluids 
to  the  same  height  as  though  the  bore  were  throughout  only  equal  to 
that  of  the  smaller  part.  As  this  experiment  does  not  succeed  in 
vacuo,  when  the  wider  end  is  immersed,  the  column  in  this  case 
is  supported  by  the  pressure  of  the  atmosphere.J 

437.  Such  are  the  leading  facts  ascertained  respecting  capil- 

*  Playfair,  Outlines  I,  177.  t  Ib.  t  Biot 

38 


298  NATURAL   PHILOSOPHY. 

\ 

lary  attraction.  Various  explanations  of  them  have  been  at- 
tempted, but  that  of  La  Place  is  most  generally  received.  Ac- 
cording to  this  high  authority,  the  action  of  the  sides  of  the  tube 
draws  up  the  film  of  fluid  nearest  to  it,  and  that  film  draws 
along  with  it  the  film  immediately  below  it,  and  so  each  film 
drags  along  with  it  the  next  below,  until  the  weight  of  the  vol- 
ume of  fluid  raised  exactly  balances  all  the  forces  which  act 
upon  it.  The  fact  that  the  elevation  of  the  water  between  the 
parallel  plates,  is  exactly  half  that  in  a  tube  of  the  same  diame- 
ter, clearly  indicates  that  the  force  resides  in  the  surrounding 
body ;  and  the  additional  fact  that  the  thickness  or  quantity  of 
that  body  makes  no  difference,  proves  that  the  force  resides  in 
the  surface,  and  that  the  action  extends  only  to  a  very  small  dis- 
tance. The  concave  surface  exhibited  by  water  and  all  fluids 
capable  of  wetting  the  tube,  (where,  of  course,  the  attraction 
between  the  fluid  and  the  tube,  is  greater  than  between  the  par- 
ticles of  the  fluid  among  themselves,)  still  further  indicates  a 
force  acting  in  the  direction  of  the  surface  of  the  tube,  while  the 
convex  surface  and  depression  of  mercury,  are  such  effects  as 
might  be  anticipated  from  the  cohesion  of  its  parts,  which  great- 
ly exceeds  its  attraction  for  glass  and  other  substances. 

438.  Various  phenomena  in  nature  and  art  are  explained  upon 
the  principles  of  capillary  attraction.     Capillary  action  is  not 
confined  to  tubes,  but  is  exerted  among  all  substances  which  are 
perforated  by  pores,  or  subdivided  by  fissures  or  interstices.     On 
this  power  depend  chiefly  the  functions  of  the  excretory  vascular 
system  in  plants  and  animals,  and  hence  also  the  ascent  of  hu- 
midity through  the  shivered  fragments  of  rocks,  unglazed  pottery, 
gravel,  earth,  and  sand.     Thus  if  the  pores  of  the  human  skin 
(which  are  known  to  be  exceedingly  small)  are  estimated  at  the 
T1S-V oth  part  of  an  inch  in  diameter,  they  will  support  the  fluids 
that  circulate  through  them  to  the  height  of  120  inches,  or  ten 
feet,  or  higher  than  is  required  for  the  animal  system.*     The  as- 
cent of  the  sap  in  trees  has  usually  been  ascribed  to  capillary 
attraction,  their  circulating  vessels  being  a  congeries  of  small 
tubes  ;  but  some  physiologists  maintain  that  this  action  is  depend- 
ent, not  on  the  mechanical  structure,  but  upon  something  which 
they  denominate  the  living  principle  of  vegetables. 

439.  According  to  Professor  Leslie,  if  a  soil  of  gravel  contains 
pores  100th  part  of  an  inch  in  diameter,  water  will  ascend  in  it 
by  capillary  action  more  than  four  inches ;  and  supposing  the 
particles  of  coarse  sand  to  have  interstices  of  the  500th  part  of 
an  inch,  the  water  would  rise  through  a  bed  of  sixteen  inches  ; 
and  if  the  pores  were  diminished  to  the  10,000th  part  of  an  inch, 
water  would  rise  twenty-five  and  a  half  feet.     Hence,  in  agri- 

»  Leslie,  Elem.  Nat.  Phil.,  Vol.  I. 


HYDRAULICS.  299 

culture,  are  derived  the  advantages  of  deep  and  perfect  tillage ; 
since,  the  more  effectually  a  soil  is  pulverized,  the  better  fitted 
it  is  to  raise  and  retain  water  near  the  surface. 

440.  Several  familiar  examples  of  capillary  attraction  may  be 
added.     A  piece  of  sponge,  or  a  lump  of  sugar,  touching  water 
by  its  lowest  corner,  soon  becomes  moistened  throughout.     The 
wick  of  a  lamp  lifts  the  oil  to  supply  the  flame,  to  the  height  of 
several  inches.     A  capillary  glass  tube,  bent  in  the  form  of  a 
syphon,  and  having  its  shorter  end  inserted  in  a  vessel  of  wa- 
ter, will  fill  itself  and  deliver  over  the  water  in  drops.     A  lock 
of  thread  or  of  candle-wick,  inserted  in  a  vessel  of  water  in  a 
similar  manner,  with  one  end  hanging  over  the  vessel,  will  ex- 
hibit the  same  result.     An  immense  weight  or  mass   may  be 
raised  through  a  small  space,  by  first  stretching  a  dry  rope  be- 
tween it  and  a  support,  and  then  wetting  the  rope.* 

RESISTANCE   OF    FLUIDS. 

441.  The  resistance  to  a  body  moving  in  a  fluid,  arises  from 
the  inertia,  from  the  cohesion,  and  from  the  friction  of  the  fluid, 
admitting  the  particles  to  be  in  contact.     The  influence  of  this 
last  cause,  granting  it  to  exist,  is  probably  very  small ;  and  the 
second  is  in  most  fluids  inconsiderable,  when  compared  with  the 
inertia.     The  resistance,  therefore,  which  we  shall  here  consider, 
is  that  which  arises  from  the  inertia  of  the  fluid,  f 

442.  The  resistance  which  a  plane  surface  meets  with  while  it 
moves  in  a  fluid,  in  a  direction  perpendicular  to  its  plane,  is  pro- 
portioned to  the  square  of  its  velocity. 

Whatever  motion  or  momentum  is  imparted  to  the  fluid,  ex- 
actly the  same  amount  is  extinguished  in  the  moving  body,  con- 
stituting the  resistance  (R.)  But  the  momentum  is  proportioned 
to  the  quantity  of  matter  and  velocity  conjointly  ;  or  Mac  QxV. 
Again,  in  the  present  case,  the  quantity  of  fluid  displaced  must 
evidently  be  proportioned  to  the  velocity  of  the  moving  body ; 
that  is,  Qx  V.  Therefore,  M  or  Roc  V2. 

This  proposition  is  found  to  hold  good  in  practice,  where  the 
velocity  is  very  small,  as  the  motions  of  boats  or  vessels  in  wa- 
ter ;  but  when  the  velocity  becomes  very  great,  as  that  of  a  can- 
non ball,  the  resistance  increases  in  a  much  higher  ratio  than  as 
the  square  of  the  velocity.  (See  Art.  359.)  Since  action  and 
reaction  are  equal,  it  makes  no  difference,  in  the  foregoing  pro- 
position, whether  we  consider  the  plane  in  motion  and  the  fluid 
at  rest,  or  the  fluid  in  motion  and  striking  against  the  plane  at 
rest. 

•  Arnott's  El.  Phys.,  I,  19— Robison's  Mech.  Phil.         t  Vince's  Hydrostatics. 


300  NATURAL    PHILOSOPHY. 

443.  On  account  of  the  rapidity  with  which  the  resistance  in- 
creases as  the  velocity  is  augmented,  when  a  vessel  or  a  steam- 
boat is  moving  in  water,  it  is  only  a  comparatively  moderate 
velocity  that  can  possibly  be  given  to  it.     A  vessel  driven  by  a 
wind  which  moves  60  miles  an  hour,  is  not  carried  forward  faster 
than  at  the  rate  of  12  or  14  miles  per  hour.     Steamboats  are 
sometimes  urged  forward  at  the  rate  of  16  miles  an  hour;  but 
to  gain  the  additional  speed  over  and  above  12  miles,  requires  a 
great  expenditure  of  force.     If  a  steam-engine  of  20  horse  pow- 
er give  a  motion  of  4  miles  an  hour,  it  would  require  one  of  180 
horse  power  to  increase  the  speed  to  12  miles  an  hour.     But,  it 
must  be  observed,  that  the  resistance  decreases  as  fast  when  the 
velocity  is  diminished,  as  it  increases  when  the  velocity  is  aug- 
mented ;  and  consequently,  that  canals  may  have  the  advantage 
over  railways,  when  heavy  articles  are  to  be  transported  by  very 
slow  motions,  although  railways,  encountering  only  the  resist- 
ance of  the  air  instead  of  water,  have  greatly  the  advantage 
when  the  motion  is  swift.*     A  cannon  ball,  on  the  other  hand, 
meets  with  so  much  resistance  on  striking  the  water  as  to  rebound. 

Reckoning  the  resistance  to  increase  only  as  the  square  of  the 
velocity,  it  follows  that  twice  the  speed  encounters  four  times  the 
resistance  ;  four  times  the  speed,  sixteen  times ;  and  ten  times 
the  speed  one  hundred  times  the  resistance.  Hence  a  body  de- 
scending freely  through  the  air  by  gravity,  for  a  great  distance, 
does  not  continue  to  be  accelerated  throughout  the  whole  dis- 
tance, but  is  finally  brought,  by  the  resistance  of  the  air,  to  a 
uniform  motion. 

Notwithstanding  the  difficulties  attending  the  mathematical 
theory  of  hydraulics,  so  much  has  already  been  done  by  the  as- 
sistance of  practical  investigations,  that  we  may  in  general,  by 
comparing  the  results  of  former  experiments  with  our  calcula- 
tions, predict  the  effect  of  any  proposed  arrangement,  without  an 
error  of  more  than  one  fifth,  or  perhaps  one  tenth,  of  the  whole, — 
a  degree  of  accuracy  fully  sufficient  for  practice.f 

FORMATION   OP    WAVES. 

444.  When  the  surface  of  water  is  pressed  upon  unequally,  in 
parts  contiguous  to  one  another,  the  columns  most  pressed  are 
shortened,  and  sink  beneath  the  natural  level  of  the  surface, 
while  those  that  are  least  pressed  are  lengthened,  and  rise  above 
that  level.     As  soon  as  the  former  columns  have  sunk  to  a  certain 
depth,  and  the  latter  have  risen  to  a  certain  height,  their  motions 
are  reversed,  and  continue  so,  until  the  columns  that  were  at  first 
most  depressed  have  become  most  elevated,  and  those  that  were 
most  elevated  have  become  most  depressed-! 

*  See  Leslie,  Nat.  Phil.,  I,  443. 

t  Dr.  Young's  Lectures  on  Nat.  Phil.,  I,  277.         *  Playfair's  Outlines,  I,  203. 


t  HYDRAULICS.  301 

445.  The  alternate  elevations  and  depressions  of  the  surface  of 
a  body  of  water,  produced  by  a  force  acting  unequally  on  the  sur- 
face, are  called  waves. 

The  water  in  the  formation  of  waves  has  a  vibratory  or  recip- 
rocating motion,  both  in  a  horizontal  and  in  a  vertical  direction, 
by  which  it  passes  from  the  columns  that  are  shortened  to  those 
that  are  lengthened,  and  returns  again  in  the  opposite  direction. 
Progressive  motion  is  not  necessary  to  undulation.* 

446.  Sir  Isaac  Newton  first  observed  the  analogy  between  the 
motions  of  waves  and  the  vibrations  of  a  column  of  water  in  a 
recurved  tube,  and  upon  this  analogy  he  founded  his  theory  of 
waves.     Let  AFGB,  (Pig.  191,)  be  a  bent  tube,  of  equal  bore 
throughout,  having  its  sides  parallel  to  each  other  and  perpendic- 
ular to  the  horizon.     Suppose  it  to  be  filled  with  water  or  any 
fluid  to  the  height  MM'.     By  any  pressure  applied  at  M',  let  the 
column  be  depressed  to  N'  and  raised  to  E  in  the  opposite  arm. 
The  pressure  being  removed,  the  longer  column  Fig.  191. 
EF  will  preponderate  and  seek  to  regain  its  ori- 
ginal level,  but  the  ascending  column  will  not 

stop  at  M',  but  on  account  of  its  inertia,  will  as- 
cend to  E',  that  is,  to  the  same  height  as  that 
from  which  it  descended  on  the  other  side.  It 
will  now  descend  again,  and  these  reciprocal 
motions  will  continue  until  destroyed  by  the 
natural  impediments  to  motion.  On  account  of 
these,  each  successive  vibration  is  shorter  than 
the  preceding,  but  all  of  them,  like  those  of  a  pendulum,  are 
performed  in  equal  times  ;  for  the  moving  force  is  obviously 
proportioned  to  the  column  EM,  that  is,  to  the  space  through 
which  the  whole  column  vibrates  ;  and  when  the  forces  are  as 
the  spaces,  the  times  are  equal. 

447.  Now  when  the  surface  of  water  is  smooth  and  at  rest,  if 
any  force  (as  the  action  of  the  wind  or  the  fall  of  a  stone)  depress 
that  surface  in  any  particular  place,  the  contiguous  water  will 
necessarily  rise  all  around  that  place.     The  water  which  has  thus 
been  elevated,  descends  soon  after  in  conseopence  of  its  gravity ; 
and  by  the  time  it  has  reached  the  original  level,  it  will  have  ac- 
quired velocity  sufficient  to  carry  it  lower  than  that  level ;  there- 
fore, it  now  acts  as  another  original  moving  force,  in  consequence 
of  which,  the  water  will  be  raised  on  both  sides  of  it.     And  for 
the  same  reason,  the  descent  of  those  elevated  parts  will  produce 
other  elevations  contiguous  to  them,  and  so  on.     Thus  the  alter- 
nate rising  and  falling  of  the  water  in  ridges,  will  expand  all 
around  the  original  place  of  motion ;  but  as  they  recede  from 

»  Playfair. 


302  NATURAL   PHILOSOPHY.  •  ., 

that  place,  so  the  ridges,  as  well  as  the  adjoining  hollows,  grow 
smaller  and  smaller  until  they  vanish.  This  diminution  of  size  is 
produced  by  three  causes,  namely,  by  the  want  of  perfect  free- 
dom of  motion  among  the  particles  of  water,  by  the  resistance 
of  the  air,  and  by  the  remoter  ridges  being  larger  in  diameter 
than  those  which  are  nearer. 

448.  From  a  variety  of  experiments  and  observations,  it  ap- 
pears that  the  utmost  force  of  the  wind  cannot  penetrate  a  great 
way  into  the  water ;  and  that  even  in  violent  storms  the  water 
of  the  sea  is  slightly  agitated  at  the  depth  of  twenty  feet  below 
the  usual  level,  and  probably  not  moved  at  all  at  the  depth  of 
thirty  feet.*     Therefore,  the  actual  displacing  of  the  water  by 
the  wind  cannot  be  supposed  to  reach  nearly  so  low  ;  and  hence 
it  would  seem  that  the  greatest  waves  could  not  be  so  very  high 
as  they  are  often  represented  by  navigators.     But  it  must  be 
observed  that  in  storms  waves  increase  to  an  enormous  size  from 
the  accumulation  of  waves  upon  waves ;  for,  as  the  wind  is  con- 
tinually blowing,  its  action  will  raise  a  wave  upon  another 
wave,  and  a  third  wave  upon  a  second,  in  the  same  manner  as 
it  raises  a  wave  upon  the  flat  surface  of  the  water.     In  fact,  at 
sea,  a  variety  of  waves  of  different  sizes  are  frequently  seen  one 
upon  the  other,  especially  while  the  wind  is  actually  blowing. 
When  it  blows  fresh,  the  tops  of  the  waves,  being  lighter  and 
thinner  than  the  other  parts,  are  impelled  forward,  btoken,  and 
turned  into  a  white  foam,  particles  of  which,  called  spray,  are 
carried  to  a  great  distance.f 

449.  While  the  depth  of  the  water  is  sufficient  to  allow  the 
oscillation  to  proceed  undisturbed,  the  waves  have  no  progressive 
motion,  and  are  kept,  each  in  its  place,  by  the  action  of  the 
waves  that  surround  it.     But  if,  by  a  rock  rising  near  to  the  sur- 
face, or  by  the  shelving  of  the  shore,  the  oscillation  is  prevented, 
of  much  retarded,  the  waves  in  the  deep  water  are  not  balanced 
by  those  in  the  shallower,  and  therefore  acquire  a  progressive 
motion  in  this  last  direction,  and  form  breakers.     Hence  it  is  that 
waves  always  break  against  the  shore,  whatever  be  the  direc- 
tion of  the  wind.     I^eakers  formed  over  a  great  extent  of  shore, 
are  distinguished  by  the  name  of  surf.     The  surf  is  greatest  in 
those  parts  of  the  earth  where  the  wind  blows  always  nearly  in 
the  same  direction.! 

*  Boyle's  Works,  Vol.  Ill,  in  Cavallo,  I,  260.  t  Cavallo.  t  Playfair. 


PNEUMATICS.  303 


PART   IV. PNEUMATICS. 


CHAPTER  I. 

OF  THE  MECHANICAL  PROPERTIES  OF  AIR. 

450.  PNEUMATICS  is  that  branch  of  Mechanics,  which  treats  of  the 
equilibrium  and  motion  of  ELASTIC  "fluids. 

Those  laws  of  equilibrium  which  are  founded  on  the  peculiar 
nature  of  fluids  arising  from  the  mobility  of  their  particles,  are 
equally  applicable  to  Hydrostatics  and  Pneumatics.  But  certain 
additional  properties  result  from  the  elasticity  of  vapors  and 
gases,  which  may  be  conveniently  considered  under  the  latter 
head. 

Vapors  are  elastic  fluids,  which  are  produced  from  liquid  or 
solid  bodies  by  the  agency  of  heat,  and  which  readily  become 
liquid  or  solid  again  on  the  application  of  cold.  Thus  steam  is 
raised  from  boiling  water,  and  is  again  easily  condensed  by  cold 
into  the  liquid  state.  Gases  are  permanently  elastic  fluids.  They 
are  never  met  with  in  nature  either  in  the  liquid  or  solid  state, 
and  it  is  only  by  means  of  extraordinary  degrees  of  cold  or  pres- 
sure, that  they  can  be  made  to  give  up  their  elasticity  and  be- 
come liquids.  Atmospheric  air  is  a  body  of  this  class  ;  and  since 
air  and  steam  are,  with  slight  exceptions,  the  only  elastic  fluids 
employed  as  mechanical  agents,  it  is  to  these,  chiefly,  that  our 
attention  will  be  devoted. 

451.  The  effects  of  HEAT  upon  all  bodies,  are  usually  treated 
of  in  Chemistry  ;  but  a  few  of  those  effects  which  are^trictly 
mechanical,  especially  such  as  are  produced  on  aeriform  fluids, 
may  be  advantageously  considered  in  this  place. 

The  most  general  mechanical  effect  of  heat  is,  expansion. 
Heat  expands  all  bodies,  whether  solid,  liquid,  or  aeriform. 
Aeriform  bodies  are  expanded  equally  by  equal  additions  of  heat.* 
The  increase  of  volume  is  continued  without  limit,  as  the  heat  is 


*  This  and  various  other  propositions  in  Pneumatics,  are  proved  by  experiment.  It 
is  supposed  that  most  of  the  instructors  who  use  this  work,  will  have  the  means  of  il- 
lustrating or  proving  the  truth  of  these  propositions,  by  the  aid  of  appropriate  appa- 
ratus. But  even  when  this  is  not  the  case,  we  conceive  that  very  little  benefit  can 
accrue  to  the  learner  from  the  bare  description  of  experiments. 


304  NATURAL   PHILOSOPHY. 

augmented.  The  elasticity  of  a  confined  portion  of  air,  as  that 
contained  in  a  close  bottle  or  flask,  for  example,  is  uniformly 
increased  by  equal  additions  of  heat.  This  is  true  of  steam, 
however,  only  when  the  vessel  is  free  from  water  ;  for,  if  steam 
is  heated  in  contact  with  water,  in  a  close  vessel,  where  new 
portions  of  steam  are  continually  added,  without  any  enlarge- 
ment of  volume,  its  density  and  elasticity  are  rapidly  increased, 
in  a  geometrical  ratio,  and  its  mechanical  force  shortly  becomes 
so  great  as  to  burst  almost  any  vessel  that  can  be  employed  to 
contain  it. 

452.  The  properties  of  air  may  be  exhibited  under  the  form  of 
a  few  simple  propositions. 

(1.)  Air  is  material. 

The  two  essential  properties  of  matter  are  extension  and  im- 
penetrability. (Art.  2.)  That  air  has  extension,  needs  no  proof. 
That  it  is  impenetrable,  or  has  the  property  of  excluding  all 
other  matter  from  the  space  which  it  occupies,  is  proved  by  ex- 
periment. Thus  if  we  depress  in  water  a  tall  jar,  or  a  tumbler, 
we  shall  find  that  the  water  rises  only  through  a  certain  part 
of  the  vessel,  to  whatever  depth  we  immerse  it ;  and  if  to  a  hol- 
low cylinder,  made  smooth  and  closed  at  the  bottom,  we  fit 
closely  a  stopper  or  solid  cylinder,  called  a  piston,  moving  freely 
in  it,  on  applying  the  piston,  no  force  will  enable  us  to  bring  it 
into  contact  with  the  bottom  of  the  cylinder,  unless  we  permit 
the  air  within  it  to  escape.  Two  other  properties  exhibited  by 
air,  likewise  indicate  that  it  is  material :  these  are  inertia,  and 
weight.  The  inertia  of  air  is  manifested  by  the  resistance  it  op- 
poses to  bodies  moving  in  it ;  as,  for  example,  an  open  umbrella 
moved  through  the  air,  in  a  direction  parallel  with  the  staff; 
and  the  weight  of  the  air  is  shown  by  the  fact  that  a  vessel,  as 
a  bottle,  from  which  the  air  has  been  withdrawn,  (by  methods 
to  be  described  hereafter,)  weighs  less  than  before.  A  vessel  of 
the  capacity  of  a  wine  quart,  weighs  about  eighteen  grains  less 
after  the  air  is  exhausted,  than  before.  One  hundred  cubic 
inches  of  air  weigh  thirty  and  a  half  grains. 

(2.)  Air  is  a  fluid. 

This  property  is  manifested  not  only  by  the  great  mobility  of 
its  parts,  but  also  by  the  distinguishing  properties  of  fluids,  (Art. 
379,)  viz.  that  any  portion  of  air  at  rest,  presses  and  is  pressed 
equally  in  all  directions  ;  and  that  a  pressure  or  blow  applied  to 
any  part,  is  propagated  through  the  whole  mass,  and  affects 
every  part  alike.  (Art.  382.) 

(3.)  Air  is  an  ELASTIC  fluid.  (Art.  84.) 

Thus,  when  an  inflated  bladder  is  compressed,  it  immediately 
restores  itself  to  its  former  situation.  Indeed,  since  air  when 
compressed  restores  itself,  or  tends  to  restore  itself,  with  the 


PNEUMATICS. 


305 


same  force  as  that  with  which  it  is  compressed,  it  is  a  perfectly 
elastic  body.*     (Art.  84.) 

452'.  The  volume  of  a  given  weight  of  air  is  inversely  as  its  com- 
pressing  force.\ 

Let  ABCD  be  a  glass  tube  open  at  A  and  closed  at  Fig.  192. 
D.  Let  quicksilver  be  poured  into  the  tube  ;  it  will 
tend  to  rise  to  the  same  height  in  both  arms  of  the 
tube,  (Art.  391,)  but  the  air  in  CD,  by  its  elasticity, 
will  resist  its  ascent,  so  that,  when  at  rest,  it  will  stand 
much  higher  in  AB  than  in  AC.  Let  it  stand  at  e 
when  the  air  in  CD  is  compressed  into  half  its  original 
bulk  above  the  quicksilver  at  a.  Then  the  column  of 
quicksilver  Ca  will  just  balance  the  equal  column  Be, 
and  the  column  ce  will  measure  the  elastic  force  of  the 
compressed  air.  Add  more  quicksilver,  and  the  col- 
umn Ca  will  rise.  Let  it  rise  to  b,  so  that  the  air  shall  be 
compressed  into  Db,  one  fourth  its  original  volume.  It 
will  be  found,  on  measuring,  that  dh  which  measures  the 
elastic  force  of  the  air  in  D6  is  just  twice  the  height 
of  ce.  Consequently,  a  double  pressure  is  required  to 
reduce  a  given  quantity  of  air  to  half  its  volume  ;  and 
in  the  same  manner  it  may  be  shown  that  three  times 
the  pressure  reduces  the  volume  to  one  third ;  and, 
universally,  that  the  volume  is  inversely  as  the  com- 
pressing force. 

Since  the  compressing  force  is  in  each  case  a  meas- 
ure of  the  elastic  force  of  the  compressed  air,  it  fol- 
lows, that  th-e  elastic  force  of  a  given  weight  of  air,  is  as 
the  compressing  force,  and  also  inversely  as  the  volume. 
And  since  the  density  of  a  given  quantity  of  matter  is 
inversely  as  its  volume,  hence,  the  elastic  force  of  a 
given  quantity  of  air  is  as  its  density. 

453.  Before  we  proceed  further,  it  is  necessary  for  the  learner 
to  be  made  acquainted  with  the  apparatus  by  which  the  mechan- 
ical properties  of  air  are  illustrated. 


THE    AIR-PUMP. 

The  Air-Pump  is  an  instrument  used  for  the  purpose  of  ex- 
hausting the  air  from  any  given  space.  Though  of  several  dif- 
ferent forms,  yet  the  most  common  construction  is  that  repre- 
sented in  Fig.  193.  The  chief  parts  are  the  plate  A,  the  barrels 

*  The  phrase  perfect  elasticity  is  used  here  in  its  technical  sense,  and  does  not  pr». 
elude  the  idea  that  the  elastic  force  of  air  is  susceptible  of  increase  and  diminutioi. 
t  This  is  commonly  called  the  Law  of  Mariotte. 
39 


306  NATURAL   PHILOSOPHY. 

Fig.  193. 


E,  E,  and  the  pipe  or  canal  CC,  leading  from  the  plate  to  the 
oarrels.  The  glass  vessels  which  are  set  upon  the  plate,  are 
called  in  general  receivers.  A  gauge  is  sometimes  employed  (as 
represented  by  D  in  the  figure)  to  indicate  the  degree  of  exhaus- 
tion ;  but  the  nature  of  this  appendage  will  be  better  understood 
nereafter.  Such  is  the  construction  of  the  air-pump  in  general ; 
but  the  importance  of  this  apparatus  entitles  it  to  a  more  minute 
description.  In  order,  then,  fully  to  understand  the  principle  of 
the  air-pump,  and  other  kinds  of  apparatus  designed  for  pro- 
ducing a  vacuum,  we  must  learn  the  construction  of  vahes,  and 
of  the  cylinder  and  piston. 

454.  A  VALVE  is  a  contrivance  which  permits  a  fluid  to  pass  in 
one  direction,  but  prevents  its  passing  in  the  opposite  direction.  The 
clapper  seen  on  the  under  side  of  a  pair  of  bellows,  is  a  familiar 
example  of  a  valve.  The  valve  employed  in  the  air-pump,  usu- 
ally consists  merely  of  a  strip  of  oiled  silk,  tied  over  a  small  ori- 
fice. The  air  by  pressing  outward  from  the  orifice  raises-the  silk, 
opens  the  valve,  and  makes  its  escape  ;  while  by  pressing  inward 
upon  the  orifice,  it  keeps  the  strip  of  silk  close  to  the  orifice,  and 
is  therefore  prevented  from  passing  in  that  direction.  The  pis- 
ton and  cylinder  are  exemplified  in  a  common  syringe.  It  con- 
sists of  a  hollow  cylinder,  or  barrel,  to  which  is  fitted  a  short 
solid  cylinder  called  the  piston,  which  is  moved  up  and  down  the 
barrel  by  means  of  a  projecting  handle  called  the  piston  rod,  and 
is  fitted  so  closely  to  the  barrel  as  to  be  air  tight.  Suppose  now 
that  the  cylinder  is  in  a  perpendicular  position,  closed  below,  but 
open  above,  and  that  the  piston  rests  on  the  bottom.  On  draw- 
ing up  the  piston,  the  air  above  it  is  lifted  out,  and  the  space 
below  it  is  a  vacuum.  If  a  small  orifice  be  made  in  the  bottom 


PNEUMATICS. 


307 


of  the  barrel,  then  as  the  piston  is  drawn  upward,  the  air  will 
flow  in  and  no  vacuum  will  be  formed  ;  and  as  the  piston  is  de- 
pressed again,  the  air  is  forced  back.  But  by  attaching  a  valve 
to  the  orifice,  we  may  admit  or  exclude  the  external  air  at  pleas- 
ure. If  the  strip  of  silk  be  tied  on  the  outside,  then,  on  drawing 
up  the  piston,  the  air  will  not  follow,  but  the  piston  will  go  up 
heavily,  since  it  lifts  up  the  entire  weight  of  the  column  of  air 
that  rests  upon  it,  (there  being  nothing  below  it  to  act  as  a  coun- 
terpoise,) and  if  the  hand  be  withdrawn  from  the  piston  rod,  the 
piston  will  descend  spontaneously.  Again,  if  the  valve  be 
placed  on  the  inside,  then  the  external  air  will  follow  the  piston 
as  it  rises,  and  no  vacuum  will  be  formed.  If  now  the  piston  be 
depressed,  the  air  cannot  be  expelled,  (since  the  valve  closes  on 
the  orifice  in  that  direction,)  and  the  piston  cannot  be  forced 
down  to  the  bottom  of  the  barrel,  unless  a  valve  is  placed  in  the 
piston  itself,  opening  outward ;  in  this  case  the  air  of  the  barrel 
may  be  expelled  by  depressing  the  piston. 

455.  We  have  been  thus  minute  in  the  description  of  the  con- 
struction of  valves,  and  of  the  cylinder  and  piston,  because  when 
these  things  are  clearly  understood,  the  learner  will  easily  com- 
prehend the  principle  of  the  air-pump,  of  the  common  house 
pump,  of  the  steam-engine,  and  of  every  other  species  of  pneu- 
matic apparatus.  Let  us  now  return  to  the  air-pump. 

Fig.  194. 


In  the  barrels,  two  pistons  play  up  and  down,  each  of  which 
is  furnished  with  a  valve  opening  upward  into  the  open  space, 
through  which  the  piston  rods  move.  Another  valve  is  placed  at 


308  NATURAL   PHILOSOPHY. 

the  bottom  of  each  barrel,  opening  into  the  barrel.  The  piston 
rods  are  indented  bars,  to  which  a  toothed  wheel  (concealed  in 
Fig.  193,  but  seen  in  Fig.  194)  is  adapted,  which,  being  turned 
backward  and  forward  by  means  of  the  winch  G,  (Fig.  193,) 
alternately  raises  and  depresses  the  two  pistons,  as  is  represented 
in  the  preceding  figure.  Suppose  now  the  receiver  to  be  placed 
on  the  plate  of  the  pump,  one  of  the  pistons  being  at  the  top, 
and  the  other  at  the  bottom  of  the  barrel.  We  turn  the  winch, 
the  piston  rises,  and  the  air  of  the  receiver  opens  the  valve  at 
the  bottom  of  the  barrel,  and  diffuses  itself  equally  through  the 
barrel  and  the  receiver.  We  turn  the  winch  in  the  opposite  di- 
rection, the  piston  descends,  compresses  the  air  in  the  barrel  be- 
fore it,  which,  as  it  cannot  go  back  into  the  receiver,  opens  the 
valve  in  the  piston  itself,  and  escapes  into  the  vacant  space  in 
which  the  arm  of  the  piston  moves.  This  process  is  repeated 
every  time  the  piston  rises  and  falls  ;  and  it  is  the  same  in  both 
barrels,  the  two  being  employed  to  accelerate  the  process  of  ex- 
haustion, and  to  facilitate  the  working  of  the  pump,  since  the 
pressure  of  the  atmosphere  on  the  descending,  counteracts  the 
effect  of  the  same  pressure  on  the  ascending  piston.* 

456.  The  exhaustion  proceeds  at  a  rate,  which  increases  in  a  geo- 
metrical ratio. 

Suppose,  for  example,  that  the  capacity  of  one  of  the  barrels 
is  just  one  ninth  part  of  that  of  the  receiver,  including  that  of 
the  pipe  which  leads  from  the  receiver  to  the  barrel.  When  the 
piston  is  first  raised  from  the  bottom  to  the  top,  the  air  which 
previously  occupied  the  receiver,  expands  so  as  to  diffuse  itself 
equally  through  the  receiver  and  barrel.  The  barrel,  therefore, 
will  contain  a  tenth  part  of  the  whole  of  the  enclosed  air,  and 
nine  tenths  will  remain  in  the  receiver.  On  depressing  the  piston, 
this  tenth  part  is  expelled  through  the  piston  valve.  On  ele- 
vating the  piston,  the  air  remaining  in  the  receiver  (which  is 
nine  tenths  of  the  original  quantity)  diffuses  itself  equally  through 
the  receiver  and  barrel,  as  before  ;  consequently  the  barrel  con- 
tains TV  of  T9o=Tfo  of  the  original  quantity,  and  T\V  remain  in 
the  receiver.  By  continuing  this  estimate,  we  should  obtain  the 
results  expressed  in  the  following  table.f 

*  The  letters  in  Fig.  194,  are  inserted  to  aid  the  learner  in  describing  the  air-pump, 
which  can  be  done  more  conveniently  from  Fig.  194  than  from  Fig.  193. 

t  The  estimate  is  made  for  a  single  barrel :  in  the  double-barreled  air-pump,  the 
rate  of  exhaustion  will  be  just  doubV>i 


PNEUMATICS. 


309 


Number  of 
strokes. 

Part  of  the  air  expelled  al  each 

Part    remaining    in    the    re- 

Whole  quantity 
expelled. 

1 

To 

To 

To 

2 

Too 

81 

Too 

Too 

3 

81 
1000 

729 

1000 

271 

Too5 

4 

729 
10,000 

6561 
10,000 

3439 
10,000 

5     • 

6561 
100,000 

59049 
100,000 

40951 
100,000 

59049 

531441 

468559 

1,000,000 

1,000,000 

1,000,000 

531441 

4782969 

5217031 

7 

10,000,000 

10,000,000 

10,000,000 

The  numbers  in  the  second  column  denote  the  rate  of  exhaus- 
tion, and  it  is  evident  that  they  compose  a  geometrical  series,  the 
constant  ratio  being  TV  Also  the  quantities  remaining  in  the 
receiver  after  each  stroke,  compose  a  similar  series,  the  ratio 
being  the  same.  After  seven  strokes,  the  quantity  remaining  in 
the  receiver  is  less  than  one  half  the  original  quantity.  If  we 
had  taken  a  smaller  receiver,  the  rate  of  exhaustion  would  have 
been  much  more  rapid.  Thus,  if  the  receiver  had  only  the  ca- 
pacity of  the  barrel,  the  series  would  have  been  |,  £,  |,  TV,  g^» 
FV»  ris'  tie'  ITS'  TOT*  »  so  that,  with  ten  strokes  of  the  piston, 
the  air  of  the  receiver  would  have  been  rarefied  more  than  one- 
thousand  times. 

As  this  series  never  terminates,  it  is  evident  that  a  complete 
exhaustion  can  never  be  effected  by  the  air-pump.  Indeed,  in 
practice,  the  vacuum  is  far  less  perfect  than  the  theory  would 
make  it  by  the  repetition  of  the  blows  of  the  piston  ;  for  when 
the  air  in  the  receiver  becomes  very  much  rarefied,  it  has  not 
elasticity  sufficient  to  raise  the  valve  at  the  bottom  of  the  barrel; 
or  even  if  that  difficulty  is  obviated  by  a  different  construction 
of  the  valve,  still  the  difficulty  of  making  the  joints  and  valves 
perfectly  air-tight,  is  such  as  to  impair  the  perfection  of  the  void. 
In  the  most  improved  air-pumps,  the  valves  are  made  of  small 
pieces  of  metal,  which  are  opened  and  closed  by  the  action  of 
the  piston  itself.  Also  to  prevent  the  corrosion  of  brass,  arising 
from  the  action  of  the  oil  employed  to  lubricate  the  parts,  in  the 
place  of  this  material,  glass  is  now  used  for  the  barrels  and  the 
plate  of  the  pump,  and  the  piston  is  made  of  steel. 

457.  By  means  of  this  instrument,  we  may  obtain  very  strik- 
ing illustrations  of  the  mechanical  properties  of  air. 

(1.)  The  pressure  of  the  air  acts  with  great  force  on  all  bodies 
at  the  surface  of  the  earth,  amounting,  as  we  shall  show  here- 
after, to  nearly  15  pounds  upon  every  square  inch,  or  more  than 
2000  pounds  upon  a  square  foot.  Upon  so  large  a  surface,  there- 
fore, as  that  of  the  human  body,  the  pressure  amounts  to  no  less 
than  13  or  14  tons;  but  being  so  uniformly  distributed  within 


310  NATURAL   PHILOSOPHY. 

and  without,  and  on  all  sides,  it  is,  when  the  air  is  at  rest, 
scarcely  perceptible.*  In  consequence  of  this  pressure,  the  air 
insinuates  itself  into  all  fluids,  and  fills  the  pores  of  all  solids  ex- 
cept the  most  dense,  as  gold  or  platina.  The  pressure  of  the  air 
diminishes  the  tendency  -of  fluids  to  pass  into  the  state  of  vapor, 
and  of  course  raises  their  boiling  point.  Warm  water,  at  a  tem- 
perature much  below  the  boiling  point,  will  be  set  a  boiling  under 
the  receiver  of  an  air-pump,  or  in  a  vacuum  formed  in  any  other 
way.  Indeed,  if  it  were  not  for  atmospheric  pressure,  water 
would  require  only  the  moderate  heat  of  72  instead  of  212  degrees 
of  heat  to  make  it  boil ;  and  the  more  volatile  fluids,  as  alcohol 
and  ether,  would  hardly  be  found  in  nature,  in  the  liquid  state. 

(2.)  The  elasticity  of  the  air  is  such,  that  the  smallest  portion 
of  it  may  be  expanded  beyond  any  known  limits,  by  removing 
the  external  pressure.  By  this  means,  a  bubble  may  be  made 
to  fill  a  very  large  space.  On  the  other  hand,  air  has  been  con- 
densed by  pressure,  until  its  density  has  been  greater  than  that 
of  water,  still  retaining  the  elastic,  invisible  state.f  In  conse- 
quence of  its  elasticity,  air  is  set  in  motion  by  the  least  disturb- 
ance of  its  equilibrium,  whether  by  condensation  or  rarefaction, 
thus  giving  rise  to  the  phenomena  of  winds. 

(3.)  Air  is  essential  to  the  support  of  combustion,  and  to  the 
respiration  of  animals  ;  and  finally,  it  is  the  principal  medium  of 
sound.  It  may  be  further  shown,  that  the  weight  of  bodies  is 
diminished  by  the  buoyancy  of  air,  (acting  on  the  same  principle 
as  water,  Art.  397,)  and  that  light  bodies  are  sustained  in  it,  in 
consequence  of  its  greater  specific  gravity,  while,  in  a  vacuum, 
bodies  of  various  densities,  as  a  guinea  and  a  feather,  fall  toward 
the  earth  with  equal  velocities. 

These  are  the  leading  truths  which  are  established  and  illus- 
trated by  means  of  the  air-pump,  which  the  learner  will  better 
comprehend  by  witnessing  the  actual  experiments,  than  by  any 
description  of  them  that  could  be  offered. 

458.  The  condensation  of  air  is  usually  effected  by  means  of 
the  Condensing  Syringe.  This  instrument  is  a  cylinder  and 
piston,  the  cylinder  having  a  valve  opening  outward,  while  the 
piston  is  without  a  valve.  The  principle  of  its  operation  will 
be  readily  understood  from  the  figure.  Near  the  top  of  the  cylin- 
der, at  E,  is  a  small  hole  in  the  side,  which  is  immediately  be- 
low the  piston,  when  this  is  drawn  up  to  the  top  of  the  cylinder. 
On  forcing  down  the  piston,  the  air  is  driven  before  it,  and 
expelled  through  the  valve  at  the  bottom.  By  connecting  a 
bottle  or  other  close  vessel  with  the  bottom,  the  air  expelled 

*  Fishes  are  sometimes  caught  at  the  depth  of  2GOO  or  2700  feet,  where  the  pres- 
sure of  the  water  is  equal  to  80  atmospheres,  or  more  than  82  tons  to  the  square  foot ; 
yet  these  fishes  are  not  injured  by  such  an  immense  weight,  or  sensibly  impeded  in 
their  motions.  (Camb.  Mech.  p.  352.)  t  Gregory,  I,  481 


PNEUMATICS. 


311 


may  be  driven  into  that,  its  return  being  pre-  Fi£-  195. 
vented  by  the  same  valve.  The  piston  being 
drawn  up  again  above  the  opening  in  the  cylin- 
der, another  similar  portion  of  air  may  be  forced 
into  the  condensing  bottle  ;  and  thus  the  process 
may  be  continued  indefinitely.  Sometimes  a 
valve,  opening  downward,  is  placed  in  the  piston 
itself,  and  then  the  orifice  at  E  is  omitted. 

The  Condensing  Fountain  is  a  bottle,  usually 
of  copper,  partly  filled  with  water,  upon  the  sur- 
face of  which  the  air  is  condensed  by  means  of 
the  condensing  syringe.  The  fluid  being  thus 
brought  under  a  strong  pressure,  it  tends  to  issue 
with  great  force  whenever  a  pipe,  that  is  inserted 
in  the  bottle,  and  extends  below  the  surface  of 
the  water,  is  opened.  The  manner  of  its  opera- 
tion may  be  clearly  understood  from  Fig.  195, 
where  D  represents  the  spout,  having  a  long  pipe  descending  into 
the  water  R,  above  which  the  air  is  condensed.  The  celebrated 
spouting  springs  of  Iceland,  called  the  Geysers,  in  which  water 
accompanied  by  large  masses  of  rock,  is  thrown  to  the  height  of 
200  feet,  arise  from  pneumatic  pressure  acting  upon  the  surface 
of  the  water  in  the  interior  of  the  earth,  the  aeriform  substance, 
whatever  it  may  be,  being  produced  by  means  of  volcanic  action. 

The  Air-Gun  is  an  instrument  in  which  condensed  air  is  sub- 
stituted as  the  moving  force  instead  of  gunpowder.  By  means 
of  a  condensing  syringe,  air  is  strongly  condensed  in  a  metallic 
ball  furnished  with  a  valve  at  the  mouth,  where  it  is  screwed  on 
the  gun  below  the  lock.  As  the  lock  is  sprung,  it  falls  upon  a 
plug,  and  forces  it  upon  the  valve,  which  instantly  opens,  and 
the  air  rushes  into  the  barrel  of  the  gun,  and  by  its  sudden  ex- 
pansion, propels  a  ball  much  in  the  same  manner  as  gunpowder 
would  do  in  its  place. 

459.  The  Diving  Bell  is  an  apparatus  employed  for  exploring 
the  depths  of  the  sea.  It  was  formerly  made  in  the  shape  of  a 
bell,  but  is  now  more  commonly  made  square  at  the  top  and  bot- 
tom, the  bottom  being  a  little  larger  than  the  top,  and  the  sides 
slightly  diverging  from  above.  The  material  is  sometimes  cast 
iron,  the  whole  machine  being  cast  in  one  piece,  and  made  very 
thick,  so  that  there  is  no  danger  either  from  leakage  or  fracture 
Sometimes  the  diving  bell  is  made  of  planks  of  two  thicknesses, 
with  sheet  lead  between  them.  In  the  top  of  the  machine  are 
placed  several  strong  glass  lenses  for  the  admission  of  light,  such 
jis  are  used  in  the  decks  of  vessels  to  illuminate  the  apartments 
T>elow.* 


•  Ed.  Phil.  Jour.  V,  8.    Amer.  Jour.  Science,  xxii,  325. 


312  NATURAL   PHILOSOPHY. 

The  diving  bell  depends  for  its  efficacy  on  that  quality  of  air 
which  is  common  to  all  material  substances,  impenetrability ; 
that  is,  the  exclusion  of  all  other  bodies  from  the  space  it  occu- 
pies. The  principle  may  be  illustrated  by  depressing  a  tumbler 
or  jar  in  water,  with  the  mouth  downward  :  it  will  be  seen  that 
the  water  will  ascend  so  far  as  to  occupy  only  a  part  of  the  ca- 
pacity of  the  vessel,  the  upper  part  being  occupied  by  air.  As 
the  diving  bell  descends  in  the  wator,  the  air  inclosed  in  it  is 
subject  to  its  pressure,  (which  increases  with  the  depth,)  and  by 
virtue  of  its  elasticity,  it  will  become  condensed  in  proportion  to 
this  pressure.  Thus  at  the  depth  of  about  thirty-four  feet,  the 
hydrostatic  pressure  will  be  equal  to  that  of  the  atmosphere, 
and  consequently,  the  air  being  under  a  pressure  equivalent  to 
that  of  two  atmospheres,  it  will  be  condensed  into  one  half  its 
original  volume.  As  the  depth  is  increased,  the  space  occupied 
by  the  air  in  the  bell  will  be  proportionally  diminished.  Seats 
are  furnished  for  the  workmen,  and  shelves  for  tools,  and  various 
other  conveniences.  Although  at  the  depth  of  thirty-four  feet, 
the  water  would  occupy  one  half  the  capacity  of  the  vessel,  and 
more  or  less  at  different  depths,  yet  by  means  of  a  forcing  pump 
or  condensing  syringe,  communicating  between  the  atmosphere 
above  and  the  machine,  through  a  pipe,  air  may  be  ,  Fig.  196. 
thrown  in  so  as  to  exclude  the  water  entirely.  By 
the  same  means  fresh  air  may  be  conveyed  to  the 
workmen,  the  portion  of  air  rendered  impure  by  respi- 
ration being  at  the  same  time  suffered  to  escape  by 
opening  a  stop-cock  in  the  top  of  the  machine.* 

460.  Before  we  proceed  to  the  consideration  of  the 
atmosphere,  it  is  necessary  for  the  learner  to  become 
acquainted  with  another  important  instrument,  the 
BAROMETER,  by  means  of  which,  as  well  as  by  means 
of  the  air-pump,  our  knowledge  of  the  atmosphere  has 
been  greatly  enlarged. 

THE    BAROMETER. 

Let  us  take  a  glass  tube,  about  three  feet  in  length, 
closed  at  one  end  and  open  at  the  other.  We  fill  the 
tube  with  quicksilver,  and  invert  it  in  a  vessel  of  the 
same  fluid.  The  column  of  quicksilver  falls  to  a  cer- 
tain height,  about  twenty-nine  or  thirty  inches,  where, 
after  vibrating  a  few  times,  it  remains  at  rest.  The 
space  in  the  tube  above  the  quicksilver  being  void  of 
air  or  any  other  substance,  it  is  of  course  a  vacuum, 
and  is  usually  denominated  the  Torricellian  vacuum, 

*  Lardner's  Pneumatics. 


PNEUMATICS.  313 

from  Torricelli,  an  Italian  philosopher,  who  first  discovered  this 
method  of  producing  a  vacuum.  Various  precautions  are  neces- 
sary, in  order  to  preserve  this  space  free  from  air  or  any  aeri- 
form substance :  when  these  precautions  are  taken,  this  vacuum 
is  one  of  the  most  complete  that  we  can  command. 

The  column  of  quicksilver  is  sustained  by  the  pressure  of  the 
atmosphere  on  the  open  mouth  of  the  tube,  which  is  immersed 
in  the  same  fluid;*  and  it  must  have  the  same  weight  with 
a  column  of  the  atmosphere  of  the  same  base,  otherwise  it  would 
not  be  in  equilibrium  with  it.  We  hence  arrive  at  an  ac- 
curate knowledge  of  the  actual  weight  and  pressure  of  the 
air,  since  it  is  equal  to  the  weight  of  a  column  of  quicksilver 
of  the  same  base,  thirty  inches  in  length.  The  weight  of 
such  a  cylinder  of  quicksilver  is  easily  ascertained.  Since  a 
cubic  inch  of  water  weighs  252.525  grains,  and  quicksilver 
is  13.57  times  heavier  than  water,  therefore  a  cubic  inch 
of  quicksilver  weighs  3426.76  grains  ;  and  30  inches  weigh 
102802.8  grains.  But  7004  grains  troy  make  one  pound  avoir- 
dupois. Therefore,  ^^=14.7  Ibs.  It  results,  that  the  pressure 
of  the  air  on  every  square  inch  of  surface  is,  as  stated  in  Art. 
457,  about  15  Ibs.,  or  more  than  2000  Ibs.  upon  a  square  foot. 
Since  different  fluids  balance  each  other  in  opposite  columns 
pressing  base  to  base,  when  their  heights  are  inversely  as  their 
specific  gravities,  (Art.  401,)  a  column  of  water  in  the  place  of 
the  mercury  would  stand  at  the  height  of  about  34  feet.  For  quick- 
silver being  13.57  times  heavier  than  water,  the  latter  column 
must  be  13.57  times  higher  than  the  other  ;  that  is,  30x13.57= 
407.1  inches=33.92  feet. 

By  observing  from  day  to  day  the  height  of  the  column  of 
quicksilver  prepared  as  above,  we  shall  find  that  it  varies  through 
a  space  of  two  or  three  inches,  showing  that  the  atmosphere 
does  not  always  exert  the  same  pressure,  but  that  a  given  col- 
umn of  air  is  sometimes  lighter  and  sometimes  heavier.  This 
instrument,  therefore,  enables  us  to  ascertain  the  relative  weight 
of  the  air  at  any  given  time,  and  hence  its  name,  barometer.^ 
For  the  purpose  of  indicating  these  variations  with  minuteness 
and  precision,  a  graduated  scale  is  attached  to  the  barometer, 
divided  into  inches  and  tenths  of  an  inch,  and  usually  extending 
from  twenty-seven  to  thirty-one  inches, — a  space  which  is  more 
than  sufficient  to  comprehend  all  the  natural  variations  in  the 
weight  of  the  atmosphere. 

*  As  young  learners  sometimes  find  a  difficulty  in  conceiving  clearly  how  the 
pressure  of  the  air  acts  in  this  case,  we  subjoin  a  remark  or  two.  It  must  be  re- 
collected, that  any  impulse  or  pressure  exerted  on  the  surface  of  the  fluid  in  the 
vessel,  extends  alike  to  all  parts  of  it,  (Art.  382  ;)  and  since  fluids  act  upward  as 
well  as  downward,  it  is  plain  that  the  pressure  acts  in  sustaining  the  column  of 
mercury  in  the  same  manner  as  though  it  were  applied  directly  to  the  mouth  of 
the  tube. 

t  From  Paoos  weight  and  /itrpov  measure. 

40 


314 


NATURAL    PHILOSOPHY. 


Fig.  197. 


-31 


461.  As  these  changes  of  weight  are  sometimes  very  minute,  a 
contrhance  called  a.  vernier  is  attached  to  the  scale,  by  means  of 
which  the  tenth  of  a  tenth,  that  is,  the  hundredth  part  of  an  inch, 
may  be  estimated.  The  vernier  consists  of  a  small  plate  movable 
up  and  down  by  a  screw  upon  the  graduated  part  of  the  barome- 
ter, and  is  divided  as  follows.  Let  AB  (Fig.  197,)  represent  the 
upper  part  of  a  barometer,  the  level  of  the  mercury  being  at  C 
namely,  at  30.3  inches,  and  nearly  anoth- 
er tenth.  The  vernier  being  brought  (by 
a  screw  which  is  usually  attached  to  it)  to 
coincide  with  the  surface  of  the  mercu- 
ry, we  look  along  down  the  scale,  until 
we  find  that  the  coincidence  is  at  the  8th 
division  of  the  vernier.  Now  as  the  ver- 
nier gains  TV  of  TV= Tf¥  of  an  inch  at 
each  division  upward,  it  of  course  gains 
T£o-  in  eight  divisions.  The  fractional 
quantity,  therefore,  is  .08  of  an  inch,  and 
the  height  of  the  mercury  is  30.38.  If 
the  divisions  of  the  vernier  were  such, 
that  each  gained  ^V  (when  60  on  the 
vernier  would  equal  61  on  the  limb)  on  a 
limb  divided  into  degrees,  we  could  at 
once  take  off  minutes  ;  and  were  the  limb 
graduated  to  minutes,  we  could  in  a  simi- 
lar way  read  off  seconds. 


-29 


Bl 


462.  When  the  barom"^  is  to  remain  stationary  in  a  single 
place,  the  cistern  contain)  the  mercury  is  made  of  the  form  of 
a  wide  basin.  In  a  vessel  of  small  diameter,  the  rise  of  the  mer- 
cury in  the  cistern,  as  it  descended  in  a  tube,  would,  by  its  reac- 
tion, tend  to  raise  the  mercury  in  the  tube,  for  which  effect  a 
correction  would  be  necessary.  But  in  a  wide  cistern,  the  differ- 
ence of  level  occasioned  by  the  rise  and  fall  of  the  mercury  in 
the  tube  is  so  small,  that  it  may  be  safely  neglected. 

But  it  is  often  desirable  to  have  the  barometer  so  constructed, 
that  it  may  be  conveniently  carried  from  place  to  place  without 
danger  of  derangement.  Portable  barometers  are  constructed  in 
several  different  ways.  In  one,  the  mercury  of  the  cistern  is  in- 
closed in  a  leathern  bag,  to  the  bottom  of  which  is  affixed  a  screw. 
On  turning  the  screw,  the  mercury  is  forced  up  the  tube  until  it 
completely  fills  it,  and  remains  fixed.  Over  the  mercury  of  the 
cistern  an  ivory  float  is  placed,  which  is  brought  by  means  of  the 
screw  to  a  mark  on  the  stem,  which  shows  when  the  mercury  is 
at  the  level  whence  the  divisions  on  the  scale  were  commenced. 

The  portable  barometer,  furnished  with  the  above,  or  some 
similar  contrivance  for  keeping  the  mercury  steady,  is  sometimes 


PNEUMATICS.  315 

made  of  the  form  of  a  walking  cane,  and  thus  becomes  very  con- 
venient for  taking  the  altitude  of  mountains.* 

463.  Since  the  variations  of  the  barometer  correspond  to  the 
variations  in  the  weight  of  the  air  at  the  same  place,  and  since 
these  variations  are  connected  with  changes  of  weather,  this  in- 
strument thus  becomes  a  weather  glass,  and  enables  us  in  certain 
cases,  to  foresee  changes  of  weather.  The  most  important  indi- 
cations of  the  barometer  are,  that  its  rise  denotes  fair,  and  its  fall 
denotes  foul  weather,  whatever  may  be  its  absolute  height.  Also, 
a  sudden  and  extraordinary  descent  of  the  mercury  attends,  and 
frequently  precedes,  a  violent  wind.  The  immediate  cause  of  the 
descent  of  the  barometer,  is  undoubtedly  a  rarefaction  of  the  air 
at  that  place  ;  but  the  cause  of  this  rarefaction  itself,  it  may  be 
difficult  to  account  for.  The  consideration  of  this  point  will  be 
resumed  hereafter. 

The  mean  pressure  of  the  atmosphere,  as  indicated  by  the  ba- 
rometer, is  nearly  the  same,  at  the  level  of  the  sea,  in  all  parts 
of  the  earth,  corresponding  very  nearly  to  30  inches  of  mercury. 
This  fact  has  been  verified  by  numberless  observations,  made 
with  the  barometer  in  both  hemispheres,  from  the  equatorial  to 
the  polar  regions.  The  following  results  for  several  places,  in 
different  latitudes,  corrected  for  temperature,  elevation  above  the 
level  of  the  sea,  and  the  influence  of  the  earth's  rotation  on  its 
axis,  are  nearly  uniform. 

Latitude.  Bar.  Pressure. 

Calcutta,       .  .  .22*  35'  .         .  29.776 

London,         .  .  .     51     31  .  29.827 

Edinburgh,  .  .  .     55     56  .  29.835 

Melville  Island,  .  .     74     30  .  29.884 

But,  though  the  mean  pressure  of  the  atmosphere  is  nearly  the 
same,  at  the  level  of  the  sea,  over  the  whole  globe,  the  extent  of 
the  variations  to  which  it  is  liable,  is  exceedingly  different  in  dif- 
ferent parallels  of  latitude.  In  the  equatorial  regions,  the  range 
of  the  barometer  is  much  more  limited  than  within  the  polar  cir- 
cles ;  and  in  the  frigid  zones,  it  is  more  limited  than  in  the  tem- 
perate. Within  the  tropics,  the  fluctuations  of  the  barometer  do 
not  much  exceed  £  of  an  inch.f  At  New  York  the  variation 
does  not  much  exceed  1£  inches,  while  in  Great  Britain  it  is  as 
great  as  3  inches.J  The  most  extensive  variations  take  place 
between  the  latitudes  of  30°  and  60°,  being  the  zone  in  which 

*  To  render  the  indications  of  the  barometer  entirely  worthy  of  confidence,  a  num. 
her  of  practical  precautions  are  requisite  in  the  mode  of  constructing  and  filling,  a. 
good  account  of  which  may  be  found  in  Renwick's  Mechanics,  p.  382.  Many  of  the 
cheaper  kinds  exposed  for  sale  are  very  inaccurate.  Daniell's  barometer  is  one  of 
the  best. 

t  Daniell's  Meteorology,  I,  108.  t  Renwick,  p.  384 


316  NATURAL   PHILOSOPHY. 

the  annual  changes  cf  temperature  and  humidity  possess  the 
widest  range.* 

The  barometer  also  undergoes  certain  variations  corresponding 
to  the  different  hours  of  the  day,  called  its  horary  variations. 
To  ascertain  the  nature  of  these  at  any  given  place,  a  long  series 
of  observations  must  be  made,  from  which  the  maximum  and 
minimum  height  maybe  deduced.  Mr.  Redfield  states  the  mean 
range  of  the  diurnal  oscillation  between  10  A.  M.  and  6  P.  M.,  to 
be  at  New  York,  .039  inches.f 

464.  Shortly  after  the  invention  of  the  barometer,  it  was  ob- 
served that  the  mercury  descends,  when  the  instrument  is  carried 
to  a  more  elevated  situation.    The  descent  is  found  to  be  about  T'T 
of  an  inch  for  87  feet.     From  this  observation,  we  may  deduce 
nearly  the  specific  gravity  of  air  compared  with  mercury  or  wa- 
ter ;  for  ~  of  an  inch  of  mercury  has,  it  appears,  the  same  weight 
as  87  feet,  or  1044  inches,  of  air.     Consequently,.!  inch  of  mer- 
cury weighs  as  much  as  10440  inches  of  air ;  that  is,  mercury  is 

10440  times,  and  water  is  I  — •  —  =•  I  769  times,  heavier  than  air. 
\13.57      / 

465.  The  learner  is  now  prepared  to  understand  the  principles 
on  which  are  constructed  the  several  gauges  used  in  connection 
with  the  air-pump,  to  indicate  the  degree  of  exhaustion. 

The  gauge  represented  at  D,  Fig.  193,  consists  of  a  glass  tube 
filled  with  mercury,  and  inverted  in  a  small  jar  of  the  same  fluid, 
and  covered  over  with  a  receiver.  This  apparatus  is  placed  upon 
the  smaller  plate  of  the  pump,  which  is  connected  with  the  larger 
plate,  by  a  horizontal  pipe.  Consequently,  when  the  air  in  the 
receiver  H  is  rarefied  by  working  the  pump,  the  air  in  the  small 
receiver  D,  being  rarefied  in  the  same  degree,  will  at  length  have 
its  elasticity  so  much  diminished,  as  to  be  unable  to  sustain  even 
the  short  column  of  mercury  in  the  tube.  The  mercury,  there- 
fore, will  descend  in  the  tube,  and  will  approach  toward  the 
level  of  the  fluid  in  the  jar,  and  will  come  nearer  to  it  in  propor- 
tion as  the  exhaustion  is  more  perfect. 

The  gauge  exhibited  in  Fig.  194,  G,  (which  is  connected  im- 
mediately with  the  receiver,)  acts  on  a  different  principle.  It 
consists  of  a  tube,  about  30  inches  long,  open  at  both  ends,  the 
lower  end  dipping  into  a  small  vessel  of  quicksilver,  and  the  up- 
per end  opening  into  the  receiver.  On  turning  the  pump,  the 
pressure  is  diminished  on  the  upper  surface  of  the  mercury  in  the 
tube,  and  the  external  pressure  of  the  atmosphere  forces  up  the 
fluid  to  a  height  corresponding  to  the  degree  of  exhaustion.  A 
scale,  graduated  intb  inches  and  tenths,  is  attached  to  the  tube. 

*  Ed.  Encyc.  Art.  '  Physical  Geography.'  t  Amer.  Jour,  xxvui,  158. 


PNEUMATICS. 


317 


The  syphon  gauge,  represented  in  Fig.  198,  is  Fig.  198. 
screwed  upon  the  small  plate  of  the  pump,  instead 
of  the  apparatus  exhibited  at  D,  Fig.  193.  Previ- 
ous to  exhaustion,  the  quicksilver  is  sustained  in 
the  arm  A  of  the  tube  by  the  atmospheric  pressure. 
When  this  pressure  is  diminished  to  a  certain  ex- 
tent, the  column  of  quicksilver  descends,  and  in  a 
perfect  exhaustion  would  attain  the  same  level  in 
both  arms  of  the  tube.  Consequently,  the  nearer  it 
approaches  to  that  level,  the  better  is  the  exhaustion. 

466.  The  elasticity  of  air  may  be  increased  ei- 
ther by  compressing  it,  or  by  heating  it  in  a  confined  state  ; 
and  its  elasticity  may  be  diminished  either  by  lessening  the 
pressure,  or  by  cooling  it.  The-  elasticity  of  springs  is  known  to 
be  frequently  impaired  by  continued  action.  This  is  not  the 
case  with  air.  Air  has  been  left  for  several  years  very  much 
compressed  in  suitable  vessels,  in  which  there  was  nothing  that 
could  have  a  chemical  action  upon  it ;  and  afterward,  on  re- 
moving the  unusual  pressure,  and  restoring  the  same  tempera- 
ture, the  air  has  been  found  to  recover  its  original  bulk,  which 
shows  that  the  continuance  of  the  pressure  had  not  diminished 
the  elasticity  of  it  in  the  least  perceptible  degree.* 


CHAPTER  II. 

OF  THE  ATMOSPHERE  AND  ITS  PHENOMENA. 

467.  THE  knowledge  now  acquired  of  the  properties  of  elastic 
fluids,  will  qualify  the  learner  to  enter  advantageously  upon  the 
study  of  the  entire  body  of  the  air,  which  constitutes  the  atmo- 
sphere.    Let  us  therefore  now  proceed  to  consider  its  weight, — 
its  extent  and  density, — and  its  relations  to  heat  and  moisture, 
giving  rise  to  the  various  phenomena  of  Meteorology. 

468.  The  WEIGHT  of  the  entire  atmosphere  may  be  easily  esti- 
mated by  means  of  the  barometer ;  for,  taking  the  medium  height 
of  the  mercury  at  thirty  inches,  the  weight  of  the  atmosphere  is 
equal  to  that  of  a  sea  of  quicksilver  covering  the  whole  earth  to 
the  depth  of  two  and  a  half  feet.     This  would  add  five  feet  to 
the  diameter  of  the  globe,  and  the  contents  of  the  whole  mass  of 
quicksilver,  in  cubic  feet,  would  be  equal  to  the  difference  be- 


*  Cavallo,  II,  p.  225. 


318  NATURAL   PHILOSOPHY. 

tween  the  solid  contents  of  the  globe,  and  those  of  a  sphere  of  a 
diameter  five  feet  greater.  Having  the  number  of  cubic  feet  of 
quicksilver,  we  have  only  to  multiply  that  number  by  the  weight 
of  one  foot  (  =13.57  x62|=848.125  Ibs.)  The  calculation  pro- 
ceeds as  follows. 

Let  R  denote  the  radius  of  the  earth  ;  r  the  height  of  the  mer- 
cury ;  -K  the  ratio  of  the  circumference  of  a  circle  to  its  diameter,  or 

4irR3 
3.14159;  the  solidity  of  the  globe=—  —  ; 

Do.  of  the  sphere,  including  the  mercury  =          —  —  ; 

o 

Do.  of  the  3 


But  since  r  denotes  but  a  very  small  fraction  of  R,  the  two  last 
terms  have  so  small  a  value,  that  they  may  be  thrown  out  with- 
out materially  affecting  the  result,  and  the  contents  of  the  mass 
of  quicksilver  will  be  4*RV.  Substituting  for  these  several 
quantities  their  numerical  values,  we  have  4(3956  x  5280)2  x 
3.14159x2.5=number  of  cubic  feet  in  the  mass  of  mercury; 
which  being  multiplied  by  848},  gives  11,624914,885408,838323 
pounds,  or  more  than  eleven  trillions  of  pounds,  or  five  thousand 
billions  of  tons.* 

Were  the  atmosphere  of  equal  density  throughout,  it  would  be 
easy  to  determine  its  height,  since  opposite  columns  of  different 
fluids  are  in  equilibrium,  when  their  heights  are  inversely  as  their 
specific  gravities.  (Art.  401.)  Therefore,  as  the  specific  gravity 
of  air  is  to  that  of  quicksilver,  so  is  the  height  of  the  column  of 
quicksilver  to  the  corresponding  height  of  the  column  of  air  that 
balances  it.  That  is,  1  :  10440  :  :  2.5  :  26100  feet  =5  miles 
nearly. 

But  the  atmosphere  is  very  far  from  being  throughout  of  uni- 
form density.  Several  causes  conspire  to  produce  this  result. 
1.  The  different  quantities  of  superincumbent  air  at  different 
altitudes  ;  2.  The  decreasing  attraction  of  the  earth  in  proportion 
as  the  square  of  the  distance  from  its  center  increases  ;  3.  The 
influence  of  heat  and  cold  ;  4.  The  admixture  of  vapors  and  other 
fluids  ;  5.  The  attraction  of  the  moon  and  other  celestial  bodies,  f 
That  the  lower  strata  of  the  atmosphere  are  far  more  dense  than 
the  upper,  will  be  obvious  from  this  consideration,  that  the 
portions  which  rest  on  the  surface  of  the  earth,  sustain  the 
weight  of  the  whole  body  of  the  atmosphere,  which,  as  appears 
above,  is  immensely  great.  But  the  density  of  air  is  as  the 
compressing  force.  (Art.  452'.)  As  we  ascend  from  the  earth, 

*  A  less  accurate  method  of  finding  the  weight,  of  the  atmosphere,  is  to  multiply 
the  number  of  square  inches  on  the  surface  of  the  globe  by  fifteen  pounds. 
1  Cavallo,  I,  227. 


PNEUMATICS.  319 

the  weight  sustained  is  constantly  diminished,  and  the  density 
lessened,  according  to  the  following  law. 

469.  The  densities  of  the  air  decrease  in  a  geometrical,  as  the 
distances  from  the  earth  increase  in  an  arithmetical  ratio. 

For,  let  us  suppose  that  the  strata  of  air  are  taken  so  thin,  that 
the  density  of  each  may  be  considered  as  uniform  throughout. 
Let  the  density  of  the  inferior  stratum  be  A,  that  of  the  second 
B,  of  the  third  C,  and  so  on.  Moreover,  let  a  be  the  weight  of 
the  whole  column  of  the  atmosphere  including  A  ;  b,  the  weight 
of  the  column  when  A  is  taken  away  ;  c,  its  weight  when  A  and 
B  are  subtracted,  and  so  on.  Then  the  weight  of  the  first  stra- 
tum is  a— b,  that  of  the  second,  b— c,  &c.  Now  the  densities  of 
two  bodies  of  the  same  volume  are  as  their  weights.  Therefore, 
A  :  B  : :  a — b  :  b—c.  But  since  the  densities  are  as  the  pressures, 
(Art.  452')  and  the  pressures  are  the  weights  of  the  incumbent 
volumes,  therefore,  A  :  B  : :  b  :  c.  Hence,  a — b  :  b—c  :  :b  :  c,  .'. 
ac—bc—b^-bc,  .'.  ac=b^,  /.  a  :b  :  :b  :  c ;  that  is,  the  weights, 
and  consequently  the  densities  of  the  successive  strata,  form  a 
geometrical  series.  If,  therefore,  at  a  certain  distance  from  the 
earth,  the  air  be  twice  as  rare  as  at  the  surface  of  the  earth,  at 
twice  that  distance  it  will  be  four  times  as  rare,  at  three  times 
that  distance  eight  times  as  rare,  &c. 

470.  By  observations  on  the  barometer  at  different  altitudes, 
aided  by  calculation,  it  is  ascertained,  that  at  the  height  of  seven 
miles  above  the  earth,  the  air  is  only  one  fourth  as  dense  as  it  is 
at  the  surface.*     Hence  if  we  take  an  arithmetical  series,  in- 
creasing by  seven,  to  denote  different  heights,  and  a  geometrical 
series  whose  constant  multiplier  is  one  fourth,  to  denote  the 
corresponding  densities,  we  may  easily  ascertain  the  density  of 
the  air  at  any  proposed  elevation. 

Arithmetical  series,  7     14     21     28       35         42  49 

Geometrical  series,  1  TV  «V  2  je  ToV*  4  oVo  TeH* 
From  this  table  it  appears,  that  at  the  height  of  twenty-one 
miles,  the  air  is  sixty-four  times  as  rare  as  at  the  surface  of  the 
earth  ;  at  the  height  of  forty-nine  miles,  sixteen  thousand  three 
hundred  and  eighty-four  times  as  rare  ;  and  if  we  pursue  the  cal- 
culation, we  shall  find  that  its  rarity  at  the  moderate  distance  of 
only  one  hundred  miles,  is  one  thousand  millions  of  times  greater 
than  at  the  earth,f  and  of  course  would  oppose  no  sensible  resist- 
ance to  bodies  revolving  in  it.  De  Luc  ascended  in  a  balloon  to 
such  a  height  that  his  barometer  fell  to  twelve  inches.  Suppos- 
ing the  barometer  at  the  surface  to  have  stood,  at  that  time,  at 
thirty  inches,  it  follows  that  he  must  have  left  three  fifths  of  the 
whole  atmosphere  below  him  ;  for  six  inches  being  one  fifth  of 

*  Cotes,  Hyd.  Lect.  p.  103.  t  Rees's  Encyc.,  Art.  "  Atmosphere  " 


320  NATURAL   PHILOSOPHY. 

thirty,  twelve  inches  must  be  two  fifths,  and  consequently  threa 
fifths  of  the  whole  must  be  below.  His  elevation  was  upward 
of  twenty  thousand  feet.* 

If  there  were  an  opening  into  the  interior  of  the  earth,  which 
would  permit  the  air  to  descend,  its  density  would  increase  in  the 
same  manner  as  it  diminishes  in  the  opposite  direction.  At  the 
depth  of  about  thirty-four  miles,  it  would  be  as  dense  as  water  ; 
at  the  depth  of  forty-eight  miles,  it  would  be  as  dense  as  quick- 
silver ;  and  at  the  depth  of  about  fifty  miles,  as  dense  as  gold. 

471.  The  foregoing  law,  however,  does  not  afford  exact  data 
for  estimating  the  density  of  the  air  at  any  given  elevation,  since 
the  density  is  affected  by  the  several  other  circumstances  men- 
tioned in  Art.  468,  which  are  not  here  taken  into  the  account. 
Since  the  force  of  attraction  diminishes  as  the  square  of  the  dis- 
tance from  the  center  of  the  earth  increases,  this  diminution  will 
occasion  a  corresponding  decrease  of  density.     However,  as  the 
force  of  attraction  will  be  very  nearly  the  same  at  such  eleva- 
tions as  the  highest  mountains,  as  at  the  general  level  of  the 
earth,  (Art.  8,)  no  allowance  is  made  on  this  account  for  baro- 
metrical measurements,  except  in  cases  when  extreme  accuracy 
is  required.     Changes  of  temperature  produce  a  much  greater 
effect,  since  heat  expands,  and  cold  contracts  the  air  ;  and  there- 
fore, in  estimating  altitudes,  the  state  of  the  thermometer  is  al- 
ways to  be  taken  into  the  account,  in  connection  with  the  height 
of  the  barometer.     Heat  and  cold  also  affect  the  height  of  the 
mercury  in  the  barometer,  independently  of  the  pressure  of  the 
atmosphere  without,  and  therefore  it  becomes  necessary  to  re- 
duce the  observations  to  a  fixed  standard  of  temperature. 

Owing  to  these  different  causes  of  irregularity  in  the  density 
of  the  air  at  different  elevations,  it  becomes  a  problem  of  much 
nicety  and  difficulty  to  obtain  accurate  measurements  of  heights, 
by  means  of  the  barometer  ;  but  the  importance  of  the  subject 
has  led  men  of  science  to  bestow  very  great  attention  upon  it. 
We  have  room  only  to  indicate  the  general  principles  on  which 
such  measurements  depend,  leaving  the  details  to  treatises  of 
greater  extent,  f 

472.  With  regard  to  the  actual  height  of  the  atmosphere  above 
the  earth,  it  is  a  point  not  easily  determined.     Efforts  have  been 
made  to  ascertain  its  height  by  means  of  the  twilight ;  but  the 
student  is  not  prepared  to  judge  of  the  accuracy  of  this  method, 
without  a  knowledge  of  Optics  and  Astronomy.     The  considera- 

*  Lardner's  Pneumatics,  Sec.  144. 

t  The  necessary  rules  for  barometrical  measurements  may  be  found  in  Robison's 
Mechanical  Philosophy,  Vol.  Ill ;  Cavallo's  El.  Nat.  Phil.  Vol.  II ;  Gregory's  Me- 
chanics,  Vol.  I ;  Renwick's  Mechanics,  p.  386 ;  and  in  most  of  the  Encyclopedias, 
under  the  article  Barometer, 


PNEUMATICS.  321 

tion  of  it,  therefore,  belongs  to  a  subsequent  part  of  our  course 
of  instruction.  We  merely  remark  here,  that  no  great  reliance 
is  placed  upon  this  method  by  those  who  are  most  competent  to 
judge  of  it. 

If  the  decreasing  densities  of  the  air  as  we  ascend  from  the 
earth  were  accurately  expressed  by  a  geometrical  series,  (Art. 
470.)  it  is  obvious  that  such  an  atmosphere  would  be  unlimited, 
since  such  a  series  would  never  end.  But  several  considerations 
render  it  probable,  that  the  atmosphere  is  bounded  by  definite 
limits.  Such  are  the  following :  (1.)  The  heavenly  bodies  move 
in  void  spaces  ;  otherwise  they  would  meet  with  resistance 
which  would  retard  their  motions,  and  the  periods  of  their  revo- 
lutions would  not  be  unalterable,  as  is  found  to  be  the  case. 
(2.)  The  expansion  of  air  is  owing  to  a  mutual  repulsion  be- 
tween its  particles.  This  force  is  diminished  as  the  particles  are 
removed  further  asunder,  by  the  enlargement  of  its  volume  ;  and 
we  may  conceive  the  repulsive  force  to  be  so  much  diminished 
at  a  certain  distance  from  the  earth,  as  to  be  counterbalanced 
by  gravity,  which  being  inversely  as  the  square  of  the  distance 
from  the  center  of  the  earth,  is  nearly  the  same  at  all  distances 
within  a  few  miles  of  the  earth's  surface.  (Art.  8.)*  (3.)  The 
condensation  produced  by  extreme  cold,  such  as  is  known  to  ex- 
ist in  the  upper  regions  of  the  atmosphere,  will  oppose  the  ex- 
pansion of  the  air,  and  counteract  its  enlargement  of  volume 
beyond  a  certain  limit. 

473.  As  we  ascend  from  the  earth,  the  temperature  of  the  air 
constantly  diminishes  until  we  arrive  at  a  region  of  frost,  the 
lower  limit  of  which  is  called  the  term  of  perpetual  congelation  ; 
by  which  is  meant  a  certain  height  above  any  place  on  the  earth 
where,  at  a  given  time,  water  begins  to  freeze.  The  heights  of 
the  term  of  congelation  for  every  parallel  of  latitude  from  the 
equator  to  the  north  pole,  have  been  computed,  partly  from  ob- 
servation, and  partly  from  the  known  mean  temperature  of  each 
parallel,  and  the  decrement  of  heat  as  we  ascend  in  the  atmo- 
sphere ;  and  the  result  is  expressed  in  the  following  table  : — 

Latitude.  Mean  height  of  the  term  Difference  for  every 

0  of  congelation  in  feet.  5  deg.  of  latitude. 

0  15577  

5  15455  .., 122 

10  15067  388 

15  14498  569 

*  This  argument  takes  it  for  granted  that  the  air  consists  of  indivisible  atoms  ;  for 
were  the  air  infinitely  divisible,  there  would  be  no  such  increase  of  distance  between 
the  particles,  and  consequently  diminution  of  repellent  force,  as  is  here  supposed. 
But  the  existence  of  such  atoms  has  been  rendered  extremely  probable,  and  the  con- 
clusions deduced  from  the  supposition  of  such  atoms,  are  found  to  accord  well  with 
experience.  (See  Wollaston  on  the  Finite  Extent  of  the  Atmosphere.— Phil.  Trans, 
for  1822.) 

41 


322  NATURAL   PHILOSOPHY. 

Latitude-  Mean  height  of  the  term  Difference  for  every 

o  of  congelation  in  feet.  5  deg.  of  latitude. 

20   13719  779 

25   13030  689 

30   11592  1438 

35   10664  928 

40   9016  1648 

45   7658  1358 

50   6260  1398 

55 4912  1348 

60   3684  1228 

65   2516  1168 

70   1557  959 

75   748  809 

80   120  628 

From  this  table  it  appears,  that  the  height  of  the  region  of 
perpetual  frost  at  the  equator  is  almost  three  miles  ;  at  the  paral- 
lel of  35°,  about  two  miles ;  and  at  the  latitude  of  54°,  about 
one  mile  ;  while  at  the  latitude  of  80°,  this  region  approaches 
very  near  to  the  earth,  and  at  the  pole  it  probably  comes  nearly 
or  quite  down  to  the  earth.  It  is  further  to  be  remarked,  that 
the  different  heights  decrease  very  slowly  as  we  recede  from  the 
equator,  until  we  reach  the  limits  of  the  torrid  zone,  when  they 
decrease  much  more  rapidly,  the  maximum  being  at  the  paral- 
lel of  40°.  The  average  difference  for  every  5  degrees  of  lati- 
tude from  30°  to  60°,  is  1318,  while  from  the  equator  to  30°,  the 
average  is  only  664,  and  from  60°  to  80°,  it  is  only  891.  Impor- 
tant meteorological  phenomena  depend  on  this  fact. 

474.  What  is  the  cause  of  the  cold  that  prevails  in  the  upper  re- 
gions of  the  atmosphere  ? 

It  is  found  by  experiment  that  radiant  heat,  like  that  of  the 
sun,  passes  through  a  transparent  medium  without  obstruction, 
and  consequently  does  not  heat  that  medium.*  Were  the  air 
perfectly  transparent,  the  heat  of  the  sun  would  scarcely  affect 
it  at  all ;  but  the  vapors,  clouds,  and  other  substances  that  di- 
minish the  transparency  of  the  atmosphere,  intercept  a  certain 
portion  of  the  sun's  rays.  In  general,  however,  the  manner  in 
which  the  air  receives  the  heat  of  the  sun  is  this  :  the  sun's  rays 
first  communicate  their  heat  to  the  surface  of  the  earth ;  the 
stratum  of  air  next  to  the  earth  imbibes  a  portion  of  this  heat 
and  rises,  while  colder  currents  descend  or  flow  in  laterally, 
which  in  turn  become  heated  and  rise.  Hence  from  the  ground, 
when  heated  by  the  sun,  a  current  of  air  is  constantly  ascend- 
ing. On  the  other  hand,  in  the  absence  of  the  sun,  the  ground 
loses  its  heat  by  radiation,  and  becomes  colder  than  the  air  im- 

*  Black's  Lectures  on  Chemistry,  Vol.  I. 


PNEUMATICS.  323 

mediately  above  it.  The  air  therefore  now  imparts  a  portion  of 
its  heat  to  the  ground,  is  condensed,  and  remains  in  contact  with 
the  ground  unless  removed,  as  is  commonly  the  case,  by  winds. 
The  atmosphere,  therefore,  is,  for  the  most  part,  heated  and 
cooled  indirectly  by  coming  in  contact  with  the  surface  of  the 
earth. 

475.  The  changes  of  temperature  induced  on  the  earth's  sur- 
face by  the  sun's  heat,  are  not  sufficient  to  rarefy  the  air  to  any 
great  extent.  A  part,  moreover,  of  the  heat  received  from  the 
earth  in  the  day  time,  is  restored  to  it  again  at  night ;  hence  the 
rarefied  portions  of  air  do  not  ascend  far  above  the  earth  until 
they  find  their  equilibrium. 

As  a  portion  of  air  rarefied  by  heat  at  the  earth's  surface  as- 
cends, the  diminishing  pressure  which  it  sustains  as  it  rises,  has 
a  tendency  to  enlarge  its  volume.  But,  on  the  other  hand,  an 
enlargement  of  volume  increases  its  capacity  for  heat,  and  low- 
ers its  temperature,  which  tends  to  condense  it.  At  a  moderate 
elevation  above  the  earth,  these  causes  operate  to  keep  the  air 
at  rest,  and  thus  the  heat  of  the  earth  is  incapable  of  raising  the 
temperature  of  the  air,  except  within  a  short  distance,  beyond 
which  the  region  of  frost  prevails,  and  the  cold  continues  to  in- 
crease, until  it  probably  reaches,  at  a  comparatively  moderate 
distance  from  the  earth,  an  extreme  intensity.* 


RELATIONS    OF   THE    ATMOSPHERE   TO   HEAT    AND    MOISTURE. 

476.  Air  is  set  in  motion  by  every  cause  which  disturbs  its 
equilibrium.     It  is  more  sensible  than  the  most  delicate  balance, 
and  moves  with  the  slightest  inequalities  of  pressure. 

Air  is  put  in  motion  by  the  least  change  of  temperature.  Heat 
rarefies  it,  and,  as  intimated  in  Art.  475,  renders  it  specifically 
lighter  than  the  neighboring  portions,  and  it  ascends,  while  cold- 
er and  denser  portions  flow  in  to  restore  the  equilibrium.  On 
the  other  hand,  if  air  be  condensed  by  cold,  it  descends,  or  flows 
off,  until  it  meets  with  air  of  the  same  density,  where  it  rests. 
These  effects  naturally  result  from  the  perfect  fluidity  and  elas- 
ticity of  this  substance. 

477.  An  illustration  of  this  principle  is  seen  in  the  manner  in 
which  air  circulates  in  the  shaft  or  pit  of  a  deep  mine.  Such  a  cir- 
culation is  kept  up  briskly,  even  amounting  sometimes  to  a  strong 
wind,  when  two  shafts  or  pits  of  unequal  heights  are  made  to 
communicate  with  each  other  by  means  of  a  horizontal  gallery 
called  a  drift.   The  earth  remains  nearly  at  the  same  temperature 

*  According  tc  Fourier,  the  temperature  of  the  planetary  spaces  is  — 58°  Fahr. 


324 


NATURAL   PHILOSOPHY. 


summer  and  winter,  while  the  ex-  Fig.  199. 

ternal  air  is  hotter  in  summer  and 
colder  in  winter,  than  that  within 
the  mine.  Now  were  the  air 
within  the  earth  and  without  of 
the  same  density,  then  the  air  of 
the  two  shafts  and  of  the  drift 
would  remain  in  equilibrio,  the 
longer  shaft  A,  being  counter- 
balanced by  the  shorter  shaft  B, 
extended  so  as  to  embrace  C,  a 
portion  of  the  external  air,  to  A 
the  same  height  as  the  column  A. 
But  suppose  it  summer;  then  the 
air  in  A,  becoming  condensed  by 
the  influence  of  the  colder  earth, 
is  rendered  specifically  heavier, 
and  overpowers  the  columns  B 
and  C,  the  latter  consisting  of  air 
more  rarefied  than  that  within  the  earth.  Hence  the  air  will 
flow  down  the  longer,  and  out  of  the  shorter  shaft ;  and  by  bring- 
ing all  parts  of  the  mine  into  the  circulation,  the  whole  interior 
will  be  ventilated.  Again,  suppose  it  winter ;  then  the  air  in 
the  longer  shaft  being  warmer  and  more  rarefied  than  the  com- 
pound column  BC,  the  latter  preponderates,  and  the  air  flows  in 
the  opposite  direction  ;  namely,  down  the  shorter  and  out  at  the 
longer  shaft.  In  spring  and  autumn,  when  the  temperature  of 
the  atmosphere  and  the  mine  are  nearly  equal,  the  miners  com- 
plain much  of  the  suffocating  state  of  the  air.* 


Hor  Drift. 


478.  The  contemplation  of  the  motions  of  the  atmosphere  on 
a  large  scale,  as  they  exist  in  nature,  leads  to  the  subject  of 
winds ;  but  we  may  see  the  same  principle  exemplified  in  chim- 
neys and  fire-places.  The  motion  of  air  in  chimneys  may  be  un- 
derstood by  considering  the  chimney  as  one  arm  of  a  bent  tube, 
while  the  external  column,  rising  to  the  same  height,  is  the  oth- 
er arm.  Then  the  tendency  to  ascend  will  equal  the  difference 
in  the  densities  of  the  columns  of  air  in  the  opposite  arms  of  the 
tube.  When  the  air  of  the  chimney  is  rarefied  by  heat  from  the 
fire-place,  the  cold  air  from  below  makes  its  passage  upward 
into  the  partial  void,  and  thus  supplies  air  to  the  fire  to  support 
its  combustion,  and  carries  up  along  with  it  the  smoke  and  va- 
pors which  proceed  from  the  fire.  The  smoke,  it  will  be  remark- 
ed, is  carried  up,  mechanically,  by  the  ascending  current  of  hot 
air ;  for  smoke  is  itself  heavier  than  air,  and  sinks  or  descends 


*  Robison's  Mechanical  Phil.,  Ill,  763. 


PNEUMATICS.  325 

when  not  thus  supported.*  The  draught  of  the  chimney,  or  the 
strength  and  velocity  of  the  ascending  current,  is  influenced  by 
several  circumstances.  (1.)  Long  chimneys  have  a  stronger 
draught  than  short  ones,  because  they  present  a  longer  column 
of  rarefied  air ;  but  they  may  be  so  long  as  to  cool  the  air  too 
much  before  it  has  reached  the  top,  in  which  case  the  smoke 
falls  by  its  greater  specific  gravity.  In  the  case  of  large  manu- 
factories, where  a  great  number  of  fires  communicate  with  the 
same  chimney,  tall  chimneys  are  used  because  they  are  easily 
kept  hot  throughout,  and  thus  a  very  strong  draught  is  maintain- 
ed. They  also  serve  the  important  purpose  of  conveying  the 
noxious  fumes  to  a  great  elevation  in  the  atmosphere.  Long 
horizontal  pipes  frequently  have  a  bad  draught,  because  they 
cool  the  smoke  before  it  reaches  the  chimney.  (2.)  A  narrow 
throat,  opening  into  a  large  pipe  or  funnel,  makes  a  strong 
draught,  because  the  velocity  of  the  ascending  current  is  thus 
increased,  it  being  in  different  parts  of  the  chimney  inversely  as 
the  area  of  the  section.  (Art.  417.)  The  throat  of  the  chimney, 
however,  must  be  wide  enough  to  admit  freely  all  the  mixed 
products  of  the  ascending  current,  including  the  rarefied  air, 
smoke,  watery  vapor,  and  so  on ;  and,  consequently,  a  wider  throat 
is  required  for  green  wood  than  for  dry,  and  least  of  all  for  an- 
thracite coal,  where  the  amount  of  volatile  substances  expelled 
from  the  fuel  is  comparatively  small.  Small  funnels,  being  more 
easily  rarefied  than  large  ones,  are,  in  general,  to  be  preferred. 
When  reduced,  however,  beyond  a  certain  limit,  they  encounter 
too  much  resistance  from  friction,  especially  when  the  surface 
is  rough.  Indeed,  friction  impedes  the  circulation  of  air  over 
any  surface,  more  than  it  does  water,  since  the  latter  fills  up  the 
inequalities  and  deposits  a  film  in  contact  with  the  surface  which 
serves  to  lubricate  it.  Anthracite  coal,  on  account  of  the  small 
vol  ume  of  gases  produced  in  its  combustion,  admits  of  a  smaller 
funnel  than  most  other  kinds  of  fuel.  A  large  funnel  is  some- 
times rendered  unfavorable  for  carrying  smoke,  on  account  of  a 
descending  current  of  cold  air  from  without,  which  meets  it. 
This  is  especially  the  case  when  the  top  of  the  funnel  is  very 
large.  Hence  a  chimney  which  is  smoky  from  this  cause  is  fre- 
quently cured  by  inserting  in  the  top  a  smaller  funnel  of  clay. 
(3.)  A  fire-place  with  a  low  front  or  breast,  has  a  strong  draught, 
because,  in  this  case,  no  air  can  enter  the  chimney,  except  such 
as  has  felt  the  influence  of  the  fire,  and  is  thus  fitted  to  keep  the 
chimney  warm;  whereas,  if  the  throat  of  the  fire-place  is  high, 
much  of  the  air  that  flows  into  it  is  cold  and  cools  the  chimney, 
and  of  course  diminishes  the  degree  of  rarefaction  in  it.  More- 

*  This  fact  is  illustrated  by  an  experiment  suggested  by  Dr.  Franklin,  viz.  by  blow- 
ing the  smoke  of  a  tobacco  pipe  through  water  in  a  tumbler.  The  smoke,  being 
cooled  by  this  process,  rests  upon  the  surface  ol  the  water. 


326  NATURAL    PHILOSOPHY. 

over,  when  the  throat  is  near  the  fire,  it  becomes  more  intensely 
heated,  and  thus  the  degree  of  rarefaction  of  the  current  of  air 
that  passes  through  it  is  augmented  and  its  velocity  increased. 
In  the  structure  of  fire-places  and  stoves,  it  is  an  important  prin- 
ciple, that  as  little  air  as  possible  should  get  into  the  flue  of  the 
chimney,  except  what  passes  through  the  fire ;  for  if  air  which 
has  not  felt  the  influence  of  the  fire,  makes  its  way  into  the 
chimney,  it  cools  the  chimney,  and  diminishes  the  draught, 
which,  other  things  being  equal,  is  always  proportioned  to  the 
difference  of  temperature  between  the  air  within  the  chimney 
and  without.  It  is  another  important  principle,  in  regard  to  the 
economy  of  fuel,  that  no  more  air  should  traverse  the  fire  than 
what  is  necessary  to  support  the  combustion.*  All  the  air  that 
passes  through  the  fire,  over  and  above  what  undergoes  decom- 
position, cools  it,  and  carries  a  portion  of  the  heat  up  chimney. 
It  is  obvious  that  the  air  of  an  apartment  must  be  denser  than 
at  the  top  of  the  chimney,  otherwise  the  current  will  flow  down- 
ward, as  is  sometimes  the  case  when  the  room  is  very  close,  and 
the  throat  of  the  fire-place  so  large  as  to  require  a  great  quan- 
tity of  air  to  fill  the  rarefied  space,  in  which  case,  the  air  of  the 
roojn  is  speedily  exhausted.  Hence  the  advantage,  in  close 
apartments,  of  small  fire-places,  or  stoves  which  require  but  a 
small  supply  of  air.f 

478'.  A  stove  constructed  a  few  years  since  by  the  author  of  this 
work,  having  been  found  to  answer  a  good  purpose,  and  having 
come  into  extensive  use,  it  is  thought  not  improper  to  devote  a 
few  words  to  the  explanation  of  its  principles.  It  is  particularly 
designed  for  anthracite  coal. 

The  stove  consists  of  two  parts,  a  furnace,  F,  and  a  radiator, 
C.  Figure  200  represents  a  front  view  of  the  apparatus,  and 
Figure  201,  a  section.  The  radiator  is  employed  to  absorb  and 
distribute  the  heat,  instead  of  the  long  pipe  that  is  frequently 
used  for  that  purpose.  It  consists  of  two  concentric  cylinders  of 
sheet  iron,  having  a  narrow  space,  aa,  not  exceeding  an  inch  in 
diameter,  between  them.  The  inner  cylinder  (usually  called  the 
air  cylinder)  terminates  below  in  a  narrow  pipe,  like  the  neck  of 
a  bottle,  which  closes  upon  an  opening  in  the  bottom  of  the  out- 
er cylinder.  Two  vertical  partitions  pass  down  between  the  two 
cylinders,  one  of  which  is  just  behind  the  pipe  that  connects  the 
furnace  to  the  radiator,  and  the  other  is  opposite  to  it,  on  the 

*  "  It  is  a  fact  which  ought  never  to  he  forgotten,  that  of  the  air  which  forces  its 
way  into  a  closed  fire-place,  that  part  only  which  comes  into  actual  contact  with  the 
burning  fuel  and  is  decomposed  by  it  in  the  process  of  combustion,  contributes  any 
thing  to  the  heat  generated  ;  and  that  all  the  rest  of  the  air  that  finds  its  way  into 
and  through  a  fire-place,  is  a  thief  that  steals  heat,  and  flies  away  with  it  up  the 
chimney." — (Rumford,  III,  172.) 

t  See  Dr.  Franklin's  Remarks  on  the  Causes  and  Cure  of  Smoky  Chimneys, 
Works,  Vol.  II,  p.  256.  Also,  Count  Rumford's  Essays,  III,  465. 


INEUMATICS. 


327 


Fig.  200. 


Fig.  201. 


other  side  of  the  air  cylinder.  These  partitions  compel  the  heated 
current  to  pass  down  in  front,  and,  'flowing  around  the  neck  at 
n,  to  ascend  in  the  rear,  and  go  out  by  a  pipe  near  the  top,  into 
the  chimney.  The  arrows  show  the  course  of  the  current. 

The  objects  of  the  inventor  were  threefold, — economy,  purity 
of  air,  and  elegance  of  form.  The  first  end  was  attained  by  ab- 
sorbing and  diffusing  the  heat  by  means  of  the  greatest  extent 
of  metallic  surface  that  could  be  exposed  under  a  given  volume. 
In  a  large  open  pipe,  there  is  a  great  loss  of  effect,  since  a  great 
portion  of  the  heated  current  that  enters  it  from  the  furnace, 
flows  through  the  central  parts  of  the  pipe  without  coming  into 
contact  with  the  absorbing  surface.  This  evil  is  remedied  by 
making  the  current  flow  into  the  narrow  space  between  two  con- 
centric cylinders,  (aa,  Fig.  201,)  by  which  means  it  is  kept  closely 
in  contact  with  the  surfaces  of  both  cylinders.  It  is  found,  by 
calculation,  that  a  flue  of  sufficient  dimensions  may  be  obtained 
to  afford  a  free  exit  to  the  smoke  and  heated  air,  when  the  space 
between  the  cylinders  is  only  an  inch  in  diameter.  Thus,  sup- 
pose the  outer  cylinder  is  12  inches  in  diameter,  and  the  inner 
or  air  cylinder,  10  inches  ;  then  the  value  of  the  flue,  compared 
with  a  pipe,  will  be  as  follows — 

Section  of  outer  cylinder,  122x.7854 

Ditto  of  inner  ditto,  102x.7854 

Difference,  (144— 100).7854=34.5576  square  inches=circular 
ring  between  the  two  cylinders.  Or,  since  the  space  between  the 
cylinders  is  bisected  by  vertical  partitions,  half  this,  or  17.2788= 
the  effective  value  of  the  flue^a  pipe  4.7  in  diameter. 

Hence  it  appears,  that  so  small  a  space  as  one  inch  between 
two  cylinders  of  twelve  and  ten  inches,  will  afford  a  flue  as  great 
as  a  pipe  of  about  4f  inches  in  diameter,  and  sufficient  therefore 
for  a  furnace  of  corresponding  dimensions.  We  thus  find  it  in 
our  power  to  transmit  the  heated  current  from  a  furnace  through 
a  space  so  narrow  as  to  be  brought  very  effectually  into  contact 
with  the  absorbing  surfaces ;  consequently,  in  passing  a  com- 
paratively small  distance,  under  such  circumstances,  the  heat  is 


328  NATURAL   PHILOSOPHY. 

as  fully  absorbed  and  transmitted  to  the  room,  as  in  traversing  a 
great  length  of  large  open  pipe.  Hence,  there  is  a  great  saving 
of  material.  But,  secondly,  another  incidental  advantage  of 
great  importance  is  obtained  by  this  construction,  namely,  great 
purity  of  air.  The  inner  cylinder  is  no  sooner  heated  than  the 
air  within  is  rarefied,  and  cold  air  flows  in  from  below  (at  n)  as 
into  a  chimney.  This  imbibes  the  heat  of  the  cylinder  itself, 
keeping  that  from  becoming  so  hot  as  to  contaminate  the  air  of 
the  room,  while  the  air  that  circulates  through  it,  flows  out  at 
the  top  into  the  room,  in  a  current  of  warm  air  of  most  agreea- 
ble temperature.  By  this  constant  circulation  of  the  air  of  the 
room  through  the  radiator,  a  mild  and  uniform  temperature  is 
maintained  throughout  the  apartment.  If  the  apparatus  is  pro- 
perly managed,  no  part  of  it  ever  becomes  so  hot  as  to  burn  the 
particles  of  vegetable  or  animal  matter,  more  or  less  of  which 
are  usually  floating  around  a  stove,  and  \vhich  when  burnt  com- 
municate an  unwholesome  effluvium  to  the  atmosphere  of  the 
room,  as  is  experienced  in  many  close  stoves  which  become 
highly  heated.  Thirdly,  by  substituting  for  a  long  pipe  travers- 
ing an  apartment,  (which  usually  presents  a  very  unseemly  ap- 
pearance,) a  symmetrical  figure  tastefully  ornamented,  the  un- 
pleasant aspect  frequently  accompanying  close  stoves  is  avoided. 

479.  But  a  much  more  extensive  operation  of  the  principles 
by  which  the  atmosphere  is  put  in  motion,  is  exhibited  to  us  by 
nature,  in  the  phenomena  of  WINDS.  Rarefaction  by  heat  and 
condensation  by  cold  are  the  chief  causes  of  winds.  Their  dis- 
tinct existence  and  modes  of  operation,  can  frequently  be  dis- 
covered ;  and,  in  cases  where  we  can  discover  neither,  we  are 
authorized  to  infer  the  presence  of  such  a  cause,  since  it  is  so 
constantly  connected  with  the  same  effects  in  very  numerous 
examples  that  daily  pass  before  our  eyes,  while  we  are  unac- 
quainted with  any  other  adequate  causes  of  the  same  phenom- 
ena. The  motion  of  the  air,  however,  producing  a  wind,  may 
be  merely  relative,  arising  from  the  motion  of  the  spectator. 
Thus  a  steamboat,  moving  at  the  rate  of  sixteen  miles  an  hour 
in  a  perfect  calm,  would  appear  to  one  on  board  to  be  facing  a 
wind,  moving  at  the  same  rate  in  the  opposite  direction  ;  or  if, 
in  the  diurnal  revolution  of  the  earth  on  its  axis,  any  point  of 
the  earth's  surface  should  move  faster  than  the  portion  of  the 
atmosphere  above  it,  a  relative  wind  in  the  opposite  direction 
would  be  the  result.  (Art.  11.)  The  direction  of  the  wind  may 
be  modified  by  various  causes,  the  actual  direction  being  the  re- 
sultant of  two  or  more  currents  which  meet  from  different  direc- 
tions, or  of  several  different  forces.* 

»  See  some  able  remarks  ol  this  subject  by  Mr.  W.  3.  Redfield,  Amer.  Jour. 
xx,  17. 


PNEUMATICS.  329 

480.  Land  and  sea  bieezes  aflbrd  a  striking  exemplification  of 
the  principle  in  question.     These  winds  prevail  in  most  maritime 
countries,  but  more  especially  in  the  islands  of  the  torrid  zone, 
blowing  off  from  the  land  at  night,  and  toward  the  land  in  the 
day  time.     If  we  place  a  hot  stone  in  a  room,  (says  Dr.  Robison,) 
and  hold  near  to  it  a  candle  just  extinguished,  we  shall  see  the 
smoke  move  toward  the  stone,  and  then  ascend  up  from   it. 
Now,  suppose  an  island  receiving  the  first  rays  of  the  sun  in  a 
perfectly  calm  morning  ;  the  ground  will  become  warm,  and 
will  rarefy  the  contiguous  air.     If  the  island  be  mountainous, 
this  effect  will  be  more  remarkable  ;  because  the  inclined  sides 
of  the  hills  will  receive  the  heat  more  directly.     The  midland 
air  will  therefore  be  most  warmed  ;  the  heated  air  will  rise,  and 
that  in  the  middle  will  rise  fastest ;  and  thus  a  current  of  air 
upward  will  begin,  which  must  be  supplied  by  air  coming  in  on 
all  sides,  to  be  heated  and  to  rise  in  its  turn  ;  and  thus  the  morn- 
ing sea  breeze  is  produced,  and  continues  all  day.     This  current 
will  frequently  be  reversed  during  the  night,  by  the  air  cooling 
and  gliding  down  the  sides  of  the  hills,  and  we  shall  then  have 
the  land  breeze.     Professor  Mitchell  has  rendered  it  probable 
that  this  current  is  performed  in  a  constant  gyration  ;  so  that 
the  air  which  flows  in  upon  the  land  by  day,  rises,  flows  out 
above,  and  returns  again  in  the  same  current ;  and  that  the  pro- 
cess is  similar  by  night,  only  the  current  is  reversed.* 

481.  The  trade  winds  afford  an  example  of  the  operation  of 
the  same  causes  on  a  still  greater  scale.     These  winds  prevail 
in  the  torrid  zone  and  a  little  beyond  it,  extending  to  nearly  30° 
on  both  sides  of  the  equator.     When  not  affected  by  local  causes, 
they  blow  constantly  at  the  same  place,  in  one  and  the  same  di- 
rection throughout  the  year.     Their  general  direction  is  from 
northeast  to  southwest  on  the  north^  side  of  the  equator,  and 
from  southeast  to  northwest  on  the  south  side  of  the  equator. 
They  owe  their  origin  to  the  combined  agency  of  two  causes, 
namely,  the  movement  of  the  air  on  either  side  of  the»equator, 
northward  or  southward  toward  the  place  of  greatest  rarefac- 
tion, and  the  westerly  tendency  arising  from  the  effect  of  the 
earth's  diurnal  rotation  on  its  axis,f  since  they  do  not  instanta- 
neously acquire  the  greater  velocity  which  the  equatorial  regions 
have  in  consequence  of  the  earth's  revolution  on  its  axis.  J     The 
duration  of  the  trade  winds  is  variously  modified  in  different 
parts  of  the  world,  but  always  in  such  a  manner,  that  they  blow 
toward  the  point  of  greatest  rarefaction,  and  receive  a  relative 
motion  from  the  effect  of  the  earth's  diurnal  rotation. 

*  American  Journal  of  Science,  Vol.  xix. 
t  See  p.  61,  Problem  3. 

i  For  a   more  extended    description  and   inquiry  into  the  causes    of  the  trade 
winds,  see  Daniell's  Meteorology,  p.  455,  and  American  Journal  of  Science,  Vol.  xbc. 

42 


330  NATURAL    PHILOSOPHY. 

482.  The  foregoing  atmospheric  phenomena  arise  chiefly  from 
the  relations  of  air  to  Heat ;  we  are  next  to  trace  a  few  of  the 
leading  phenomena,  which  result  from  the  relations  of  air  to 
Moisture. 

By  the  action  of  the  sun's  heat  upon  the  surface  of  the  earth, 
whether  land  or  water,  immense  quantities  of  vapor  are  raised 
into  the  atmosphere,  supplying  materials  for  all  the  water  that 
is  deposited  again  in  the  various  forms  of  dew,  fog,  rain,  snow, 
and  hail.  Our  limits  will  not  allow  us  to  enter  largely  into  Me- 
teorology, under  which  head  the  various  phenomena  of  the  at- 
mosphere are  included,  but  we  shall  be  able  barely  to  glance  at 
the  subject. 

483.  A  view  of  the  constitution  of  the  atmosphere  first  propo- 
sed by  Mr.  Dalton,  a  distinguished  English  meteorologist,  now 
generally  prevails.     It  maintains,  that  the  different  aeriform  sub- 
stances of  which  the  atmosphere  consists,  namely,  oxygen,  nitro- 
gen, carbonic  acid,  and  watery  vapor,  do  not  combine  with  one 
another,  but  co-exist  as  so  many  independent  bodies ;  that  the 
watery  vapor  in  the  atmosphere,  is  raised  into  it  and  maintained 
there,  not  by  any  force  of  attraction  existing  between  that  and 
the  other  elements,  but  simply  by  the  agency  of  heat ;  and  that 
it  is  precipitated,  or  returns  to  the  state  of  water,  merely  by  the 
reduction  of  temperature,  or  the  application  of  cold.     The  quan- 
tity of  vapor  which  can  exist  in  the  atmosphere  at  any  given 
time,  depends  upon  its  elasticity,  and  that  depends  upon  its  tem- 
perature.    The  elasticity  of  vapor  increases  very  rapidly  as  we 
heat  it.     At  32°,  the  quantity  of  vapor  that  can  exist  in  the  at- 
mosphere is  only  Tj^th  of  its  volume  ;  while  at  93°,  the  extreme 
heat  of  summer,  the  quantity  rises  to  -£-$.*     The  increase  of 
quantity,  as  the  temperature  is  raised,  is  very  slow  in  the  lower 
degrees  of  the  scale,  but  very  rapid  in  the  higher ;  so  that  when 
the  air,  in  a  hot  summer  clay,  rises  from  80°  to  90°,  the  amount 
of  water  is  increased  vastly  more  than  it  would  be  by  rising  in 
the  winter  through  an  equal  number  of  degrees,  as  from  40°  to  50°. 
Hence,  hot  air  contains,  in  fact,  a  far  greater  amount  of  water 
than  cold  air,  but  being  in  the  elastic  invisible  state,  it  is  not  ob- 
vious to  the  senses,  but  such  air  appears  very  dry.     On  the  other 
hand,  when  air  which  is  very  hot,  cools  a  few  degrees,  as  from 
90°  to  80°,  the  amount  of  water  precipitated  is  much  greater 
than  is  occasioned  by  a  similar  reduction  of  temperature  in  the 
lower  degrees  of  the  scale,  as  from  40°  to  30°.     The  tempera- 
ture at  which  the  vapor  of  the  atmosphere  is  condensed  at  any 
given  time  is  called  the  dew  point.     Thus,  if  we  place  a  tum- 
bler of  cold  water  on  the  table,  in  a  warm  day,  when  the  ther- 
mometer is  at  80°,  with  a  thermometer  inserted  in  the  tumbler, 

*  Thomson,  on  Heat  and  Electricity,  p.  251. 


PNEUMATICS.  331 

and  find  that  when  the  mercury  has  sunk  to  74°  moisture 
just  begins  to  form  on  the  outside  of  the  tumbler,  then  we  say, 
the  dew  point,  for  that  time,  is  74°. 

484.  Dew  is  formed  when  the  air  comes  in  contact  with  a  sur- 
face in  a  certain  degree  colder  than  itself*     This  is  the  simplest 
deposition  of  moisture  from  the  atmosphere.    Thus  dew  is  formed 
copiously  on  a  cup  of  cold  water  during  summer,  particularly  be- 
fore a  thunder  shower ;  because  then  the  air  is  hot,  and  saturated 
with  moisture,  a  portion  of  which  it  deposits  as  soon  as  it  is  cool- 
ed.    It  is  ascertained  by  actual  observation  that  on  those  nights 
when  copious  dews  occur,  the  ground  becomes  twelve  or  four- 
teen degrees  colder  than  the  air  a  few  feet  above  it.t     Conse- 
quently, whenever  the  air,  by  circulating  over  the  surface  of  the 
ground,  comes  in  contact  with  this  colder  surface,  it  deposits  a 
portion  of  moisture  upon  it.     The  quantity  actually  deposited 
will  of  course  be  greater  as  the  difference  of  temperatures  be- 
tween the  air  and  the  ground  is  greater,  and  as  the  air  contains 
more  moisture. 

Dew  is  found  to  be  deposited  on  different  substances  unequally, 
— more  on  vegetables  than  on  dry  sand  ;  very  little  on  bright 
metallic  surfaces ;  and  none  at  all  on  large  bodies  of  water,  as 
the  ocean.  In  all  cases,  however,  these  surfaces  are  observed  to 
maintain  a  corresponding  difference  in  the  temperature  they  ac- 
quire, some  growing  much  colder  than  others  equally  exposed, 
while  the  surface  of  the  ocean  remains  at  the  same  temperature 
as  the  air  incumbent  upon  it.  The  air  therefore  is  not  cooled 
by  circulating  upon  it,  and  no  dew  is  deposited.J 

485.  Fogs  are  produced  by  watery  vapor  coming  in  contact  with 
air  colder  than  itself. 

The  vapor  may  be  such  as  is  just  rising  from  the  ground,  or 
such  as  before  existed  in  a  body  of  common  air  that  meets  and 
mixes  with  the  colder  air.  Thus,  in  a  cold  morning,  smoke  pro- 
ceeds from  various  moist  substances,  as  from  th%  breath  of  ani- 
mals, from  a  hole  in  the  ice  of  a  river,  from  wells,  and  from  many 
other  sources.  In  each  case,  the  vapor  meets  with  cold  air,  is 
condensed,  and  deposited  in  the  form  of  fog.  A  striking  exam- 
ple of  fogs  is  seen  over  rivers,  particularly  in  a  summer  morning, 
marking  out  their  courses  for  a  great  distance.  Here,  since  the 
temperature  of  the  water  changes  but  little  during  the  night, 
while  the  neighboring  land,  and  of  course  the  air  over  the  land, 
has  become  cold,  the  vapor  which  rises  from  the  river  during  the 


*  It  will  be  remarked  that  dew  is  deposited  only  at  the  surfaces  of  bodies,  and  not, 
like  fog  and  rain,  in  the  atmosphere  itself. 
t  Wells,  o»  Dew. 
\  Ibid. 


332  NATURAL   PHILOSOPHY 

night,  and  meets  with  cold  air,  is  condensed  into  fog.  The  fogs 
formed  over  shoals  and  sand  banks,  as  the  banks  of  Newfound- 
land, are  deposited  from  the  warm  and  humid  air  of  the  ocean, 
which  is  cooled  by  mixing  with  the  cold  air  over  the  banks. 
Fogs  are  phenomena  of  cold  climates,  and  are  not  so  common  in 
hot  countries ;  the  vapor  in  the  latter  situations  having  too  great 
a  degree  of  elasticity  to  permit  it  to  condense  into  a  fog  near 
the  surface  of  the  earth. 

486.  Clouds  are  dependent  on  the  same  principles  as  fogs,  con- 
sisting of  vapor  condensed  by  the  cold  of  the  upper  regions.     They 
are  formed  over  water  or  moist  places,  by  vapor  rising  so  high, 
as  to  reach  a  degree  of  cold  sufficient  to  condense  it ;  or  they 
result  from  the  mixture  of  warmer  with  colder  air,  proceeding 
always  from  the  warmer  portion. 

487.  Rain  is  produced  by  the  sudden  cooling  of  air,  containing 
large  quantities  of  watery  vapor. 

Suppose  two  bodies  of  air,  a  hotter  and  a  colder  portion,  both 
near  the  dew  point,  (Art.  483,)  to  meet ;  the  compound  would 
assume  a  temperature  which  was  the  mean  between  the  two ; 
but  the  elasticity  which  the  colder  portion  of  air  would  gain, 
would  not  equal  that  which  the  warmer  portion  would  lose,  by 
the  loss  of  the  same  amount  of  heat.  (Art.  483.)  Hence  the  elas- 
ticity of  the  mixture  would  be  less  than  the  average  elasticities  of 
the  separate  portions,  and  consequently  water  would  be  deposited. 
If  the  separate  portions  of  air  are  not  near  the  point  of  con- 
densation, still  the  elasticity  of  the  vapor  in  the  mixture  may  be 
so  much  less  than  that  of  the  constituents,  as  to  render  it  unable 
to  hold  in  the  invisible  state  all  the  water  they  contained  ;  and 
in  this  case,  more  or  less  water  would  be  deposited. 

488.  This  view  of  the  general  cause  of  rain,  (which  is  com- 
monly called  Button's  theory  of  rain,  from  Dr.  Hutton  of  Edin- 
burgh, who  fir|f  proposed  it,)  is  capable  of  being  confirmed  by  a 
extensive  induction  of  facts,  by  which  it  would  appear,  that  vari- 
able winds,  favorable  to  the  mixture  of  air  of  different  tempera- 
tures, are  accompanied  by  rain,  while  constant  winds  are  accom- 
panied by  dry  weather. 

489.  Hail  is  produced  by  the  mixture  of  exceedingly  cold  air, 
with  a  body  of  hot  and  humid  air.*     The  cold  wind  is  supposed 
to  be  derived  from  an  elevation  considerably  above  the  term  of 
perpetual  congelation,  and  to  be  suddenly  transferred  to  a  body 
of  hot  and  humid  air,  from  which  it  precipitates  the  hail.     Or  it 

«  See  some  remarks  on  Hail  Storms,  by  the  compiler  of  this  work,  in  the  Am. 
Jour.  Science,  Vol.  xvin. 


PNEUMATICS.  333 

may  be  supposed  to  result  from  a  hot  wind  blowing  from  the  tor- 
rid regions  into  the  limits  of  perpetual  frost,  and  thus  having  its 
watery  vapor  suddenly  congealed.  But  probably  the  most  fre- 
quent mode  by  which  violent  hail  storms  are  actually  produced, 
is  by  the  sudden  transportation  of  a  body  of  very  hot  and  humid 
air  to  a  great  height  in  the  atmosphere,  as  by  whirlwinds.  All 
that  the  theory  requires,  in  order  that  hail  should  be  precipitated, 
is,  that  very  hot  and  very  cold  bodies  of  air  should  be  mixed  in 
any  way  whatsoever.  Accordingly,  hail  is  found  to  be  most  fre- 
quent and  violent  in  those  regions  where  hot  and  cold  bodies  of 
air  are  most  easily  mixed.  Such  mixtures  are  rarely  formed  in 
the  torrid  zone,  since  there  the  portion  of  cold  air  would  be  want- 
ing ;  and  a  similar  difficulty  exists  in  the  frigid  zone,  for  there 
the  hot  air  is  wanting ;  but  in  the  temperate  climates,  the  heated 
air  of  the  south,  and  the  intensely  cold  winds  of  the  north,  may 
be  much  more  easily  brought  together ;  and  accordingly,  in  the 
temperate  zones  it  is  that  hail  storms  chiefly  occur.  Even  in 
these  climates,  they  are  most  frequently  found  in  places  where 
such  mixtures  are  most  easily  formed,  as  in  the  south  of  France, 
lying,  as  it  does,  between  the  Pyrenees  and  the  Alps,  which  are 
covered  with  perpetual  snows,  while  the  intervening  country  is 
subject  to  become  highly  heated  by  the  summer's  sun,  or  is  even 
visited,  especially  at  a  certain  elevation,  by  occasional  blasts  of 
the  hot  winds  that  cross  the  Mediterranean. 

490.  Mr.  Redfield  has  investigated,  with  great  success,  the 
phenomena  of  violent  storms,  especially  of  Atlantic  hurricanes, 
and  has  shown  that  they  are  generally,  if  not  always,  great 
whirlwinds.  They  usually  take  their  rise  in  the  equatorial  re- 
gion eastward  of  the  West  India  Islands ;  they  spin  like  a  top, 
(or  more  like  those  little  whirls  which  we  sometimes  observe  to 
take  up  leaves  and  other  light  substances,)  advancing  slowly  to 
the  northwest,  until  they  approach  the  coast  of  the  United  States 
near  the  latitude  of  30°,  and  then  veer  to  the  northeast,  running 
nearly  parallel  to  the  American  coast,  and  finally  spend  them- 
selves in  the  northern  Atlantic.  It  is  a  remarkable  fact,  that 
their  rotary  motion  is  always  in  one  direction,  namely,  from  right 
to  left,  or  against  the  sun.  This  motion  is  also  far  more  violent, 
especially  in  the  central  parts  of  the  storm,  than  the  progressive 
motion.  The  rotary  motion  may  amount  to  200  miles  or  more 
per  hour,  while  the  forward  motion  of  the  storm  is  not  more  than 
20  or  30  miles.  The  able  and  ingenious  papers  of  Mr.  Redfield 
on  this  subject  may  be  found  in  the  different  volumes  of  the 
American  Journal  of  Science. 

Mr.  Espy  does  not  admit  the  rotary  motion  of  storms  sup- 
posed by  Mr.  Redfield,  but  maintains  that  the  winds  blow  from  all 
directions  toward  the  center  of  the  storm ;  that  the  air  ascends  in 
the  center  in  a  continual  column ;  that  being  cooled  as  it  rises, 


334  NATURAL   PHILOSOPHY. 

/ 

its  vapor  condenses,  forming  cloud ;  that  the  latent  heat  given 
out  on  condensation  expands  the  ascending  column,  and  causes 
it  to  rise  still  higher,  condensing  more  vapor ;  and  finally,  that 
when  the  original  body  of  air  is  hot  and  largely  charged  with 
vapor,  and  rises  rapidly  to  the  higher  and  colder  regions  of  the 
atmosphere,  the  effects  are  proportionally  violent,  giving  rise  to 
tornadoes,  thunder  storms,  and  water-spouts.* 


CHAPTER  III. 

OF  THE  MECHANICAL  AGENCIES  OF  AIR  AND  STEAM. 

491.  IN  consequence  of  our  power  of  forming  a  vacuum,  either 
by  the  exhaustion  of  air*  or  by  the  condensation  of  steam,  and 
of  directing  the  force  with  which  these  elastic  substances  rush 
into  a  void  or  press  toward  it,  air  and  steam  become  important 
agents  or  prime  movers,  in  various  kinds  of  machinery.     Many 
of  the  most  useful  machines  involve  in  their  construction  the 
principles  of  both  hydraulics  and  pneumatics,  and  therefore  we 
have  reserved  an  account  of  such  machines  to  the  present  sec- 
tion. 

492.  THE  SYPHON. — If  a  tube  having  two  Fig. 
arms,  a  longer  and  a  shorter,  be  filled  with 

water,  or  any  other  liquid,  and  the  mouth  of 
the  shorter  arm  be  immersed  in  water,  the 
fluid  will  run  out  through  the  longer  arm 
until  the  whole  contents  of  the  vessel  are 
discharged.  Such  a  tube  is  called  a  syphon. 
It  may  be  filled  with  the  fluid,  either  by  suc- 
tion or  by  pouring  water  into  it,  keeping  the 
two  orifices  closed  until  the  shorter  arm  is 
immersed.  Or,  when  the  syphon  is  large, 
each  orifice  is  plugged,  and  water  is  poured  in  through  an  open- 
ing in  the  top  of  the  bend.  The  opening  being  closed,  the  short- 
er leg  is  placed  in  the  cistern,  the  plugs  removed,  and  the  fluid 
is  discharged  through  the  longer  leg.  The  principle  of  the  sy- 
phon is  as  follows.  The  atmosphere  presses  equally  on  the 
mouths  of  both  arms  of  the  tube ;  but  this  pressure  on  each  ori- 
fice is  diminished  by  the  weight  of  the  column  of  water  in  the 
leg  nearest  to  it ;  consequently,  more  of  the  atmospheric  pres- 
sure is  overcome  by  the  longer  than  by  the  shorter  column,  and 

*  Espy  on  the  Philosophy  of  Storms. 


PNEUMATICS.  335 

therefore  the  effective  pressure,  (or  what  remains,)  is  less  at  the 
mouth  of  the  longer  than  at  that  of  the  shorter  column,  and  the 
fluid  runs  in  that  direction  in  which  the  resistance  is  least;  and 
the  constant  pressure  of  the  atmosphere  on  the  surface  of  the 
fluid,  causes  the  fluid  to  continue  running  until  the  surface  has 
descended  to  the  level  of  the  mouth  of  the  syphon.  All  this  will 
be  obvious  by  inspecting  the  figure.* 

Were  the  shorter  column  thirty-four  feet  in  height,  it  would 
counterbalance  the  entire  pressure  of  the  atmosphere  on  the 
surface  of  the  fluid,  and  consequently,  there  would  be  no  force 
remaining  to  drive  the  water  forward  through  the  tube.  The 
syphon,  therefore,  can  never  raise  water  to  a  greater  height 
than  thirty-four  feet,  nor  quicksilver  higher  than  about  thirty 
inches.  It  is  obvious,  also,  that  the  place  of  delivery,  that  is, 
the  mouth  of  the  longer  arm,  must  be  at  a  lower  level  than  the 
surface  of  the  water  in  the  reservoir ;  so  that  this  instrument 
cannot  be  used  for  elevating,  but  only  for  decanting  fluids,  or 
transferring  them  from  one  vessel  to  another.  Its  chief  use  is  by 
grocers,  in  transferring  liquors  from  cask  to  cask.  It  is,  how- 
ever, sometimes  employed  to  convey  water  from  a  well  situated 
on  rising  ground,  to  a  lower  situation,  or  to  carry  it  over  a  hill  to 
a  lower  level  on  the  other  side. 

493.  Intermitting  Springs',  or  springs  which  flow  freely  for  a 
time,  and  then  cease  for  a  certain  interval,  when  they  flow  again, 


are  explained  on  the  principle  of  the  syphon.     The  annexed  cut 
represents  a  reservoir  or  hollow  in  the  interior  of  a  hill,  having 

*  We  prefer  to  describe  such  instruments  in  general  terms,  but  the  student  will  find 
it  convenient  to  recite  the  explanation  from  the  figure,  and  letters  are  annexed  to  the 
figures  for  that,  purpose. 


NATURAL,   PHILOSOPHY. 


a  syphon-shaped  outlet.  It  is  obvious,  upon  hydrostatic  princi- 
ples, that  no  water  will  be  discharged  until  the  fluid  has  reach- 
ed a  level  in  the  reservoir  equal  to  the  top  of  the  bend  in  the  out- 
let. Then  it  will  begin  to  run  out,  and  will  continue  to  run,  un- 
til the  water  has  descended  to  the  level  of  the  outlet ;  after 
which,  no  more  water  will  be  discharged  until  enough  has  col- 
lected to  reach  the  higher  level,  as  before.* 

494.  THE  COMMON  SUCTION  PUMP. — This  pump  Fig.  204. 
consists  of  two  hollow  cylinders,  placed  one  under 
the  other,  and  communicating  by  a  valve  which 
opens  upward.  The  lower  cylinder  (which  has  its 
lower  orifice  under  water)  is  called  the  suction  tube. 
In  the  upper  cylinder,  a  piston  moves  up  and  down 
from  the  bottom  to  a  spout  in  the  side  near  the  top. 
This  cylinder  we  call  the  exhausting  tube.  Suppose, 
at  the  commencement  of  the  operation,  the  piston 
is  at  the  bottom  of  the  exhausting  tube,  in  close 
contact  with  the  valve.  On  raising  it,  the  air  in 
the  suction  tube  having  nothing  to  resist  its  upward 
pressure,  lifts  the  valve  and  expands,  so  as  to  fill 
up  the  void  space,  which  would  otherwise  be  left 
in  the  lower  part  of  the  exhausting  tube.  By  this 
means,  the  air  in  the  suction  tube  is  rarefied,  and 
no  longer  being  a  counterpoise  to  the  pressure  of 
the  atmosphere  on  the  surface  of  the  well,  the  lat- 
ter preponderates  and  forces  the  water  up  the  tube, 
until  enough  has  been  raised  exactly  to  counter- 
balance the  excess  of  the  elasticity  of  the  external 
air  above  that  of  the  tube.  As  the  piston  descends, 
the  air  below  it  is  prevented  from  returning  into 
the  suction  pipe  by  the  valve  which  closes  on  its 
mouth,  but  escapes  through  a  valve  in  the  piston  itself  opening 
upward  in  the  same  manner  as  in  the  barrels  of  the  air-pump. 
The  piston  being  raised  again,  the  column  of  water  ascends  still 
higher,  until  it  makes  its  way  through  the  valve  into  the  ex- 
hausting pipe.  Then  as  the  piston  descends,  the  water  opens  its 
valve,  and  gets  above  the  piston,  and  is  lifted  to  the  level  of  the 
spout,  where  it  is  discharged.! 

The  principle  of  the  suction  pump  may  therefore  be  thus 
enunciated : 

The  water  is  raised  into  the  exhausting  pipe  by  the  pressure  of 
the  atmosphere,  and  thence  lifted  to  the  level  of  the  spout  by  means 
of  the  piston. 

*  Cavallo,  I,  387.— Library  of  Useful  Knowledge. 

t  The  student  is  requested  to  describe  from  the  figure.  It  is  recommended  to  him, 
however,  to  form  as  distinct  an  idea  as  possible  of  the  principle  of  a  machine  from  the 
general  description,  before  he  resorts  to  the  figure. 


PNEUMATICS.  337 

Since  a  column  of  water  thirty- four  feet  in  height,  in  the  suc- 
tion tube,  would  counterbalance  the  entire  pressure  of  the  atmo- 
sphere on  the  surface  of  the  well,  no  force  would  remain  to  urge 
the  column  any  higher,  and  therefore  the  valve  at  the  top  of  the 
suction  tube,  must  be  less  than  thirty-four  feet  above  the  well. 

495.  Let  us  now  consider  the  force  which  is  required  in  each 
stage  of  the  process,  to  elevate  the  piston,  exclusive  of  the 
weight  of  the  piston,  rods,  and  the  effects  of  the  friction.  Let 
the  piston  be  at  V,  and  the  level  of  the  water  in  the  suction  pipe 
at  H.  Let  the  number  of  feet  in  CH  be  called  h.  The  elastic 
force  of  the  air  in  BH  will  then  be  such  as  to  exert  a  pressure 
on  every  square  inch,  equal  to  the  weight  of  a  column  of  water, 
whose  base  is  a  square  inch,  and  whose  height,  expressed  in  feet, 
is  34— h.  In  its  ascent,  therefore,  each  square  inch  of  the  sec- 
tion of  the  piston  is  pressed  upward  by  this  force.  It  is,  on  the 
other  hand,  pressed  downward  by  the  whole  force  of  the  atmo- 
sphere, which  is  equal  to  the  weight  of  a  column  of  water  of  the 
same  base,  and  thirty-four  feet  high.  The  effective  force  then 
which  resists  the  ascent  of  the  piston,  for  every  square  inch,  is 
the  weight  of  a  column  of  water,  whose  base  is  a  square  inch, 
and  whose  height  is  the  difference  between  thirty-four  feet  and 
34— h  feet ;  that  is,  the  effective  force  is  h  feet.  Thus  it  ap- 
pears, that  it  requires  a  force  to  lift  the  piston  exactly  equal  to 
the  weight  of  a  column  of  water,  whose  base  is  equal  to  the  sec- 
tion of  the  piston,  and  whose  height  is  that  of  the  water  in  the 
suction  pipe,  above  the  level  of  the  water  in  the  well.  It  fol- 
lows, therefore,  that  as  the  water  rises  in  the  suction  pipe,  the 
force  required  to  lift  the  piston  is  proportionally  increased. 

Let  us  next  consider  the  force  required  to  lift  the  piston  in  the 
second  part  of  the  process  ;  viz.  when  the  water  raised  has 
passed  through  the  piston-valve. 

Let  the  piston  be  at  V,  and  the  level  of  the  water  at  H' ;  the 
downward  pressure  sustained  by  the  piston,  in  this  case,  is  evi- 
dently the  weight  of  the  incumbent  water  BH',  together  with  the 
weight  of  the  atmosphere.  Let  h  be  the  number  of  feet  in  the 
height  BH',  and  34+A  will  express  the  number  of  feet  in  a  col- 
umn of  water,  whose  base  is  equal  to  a  section  of  the  piston, 
and  whose  weight  is  equal  to  the  whole  downward  pressure  sus- 
tained by  the  piston. 

On  the  other  hand,  the  upward  pressure  is  produced  by  the 
weight  of  the  atmosphere  pressing  on  the  water  in  the  reservoir, 
and  transmitted  through  the  column  CB,  to  the  lower  surface  of 
the  piston.  But  as  this  pressure  has  to  support  the  column  BC, 
we  must  subtract  from  it  the  weight  of  this  column,  in  order  to 
obtain  the  effective  upward  pressure  on  the  piston.  From  a 
column  of  water  thirty- four  feet  in  height,  and  with  a  base  equal 
to  the  section  of  the  piston,  subtract  as  many  feet  as  there  are 

43 


338  NATURAL  PHILOSOPHY. 

in  BC,  and  we  shall  obtain  a  column  whose  weight  in  equal  to 
the  upward  pressure. 

The  downward  pressure  equals  34+7* 

The  upward  do.        do.     34— BC 

Remainder  7t+BC 
But  A+BC=H'B+BC=H'C. 

.  Thus  it  appears,  that  the  force  necessary  to  lift  the  piston,  i 
the  weight  of  a  column  of  water,  whose  height  is  that  of  the 
column  above  the  level  of  the  water  in  the  well,  and  whose  base 
is  equal  to  the  section  of  the  piston.  This  force,  therefore,  from 
the  commencement  of  the  process,  continually  increases,  until 
the  level  of  the  water  rises  to  the  discharging  spout,  and  thence- 
forward remains  uniform.* 

496.  From  the  foregoing  remarks,  it  is  evident  that  the  same 
force  is  expended  in  raising  water  by  means  of  the  pressure  of 
the  atmosphere,  as  when  the  force  is  applied  directly.     We  lift 
upon  the  atmosphere,  instead  of  lifting  directly  upon  the  column 
of  water.     This  method  of  raising  water  from  a  well,  is  frequent- 
ly more  convenient  than  by  a  simple  bucket,  but  the  expenditure 
of  force  is  the  same  in  both  cases. 

To  compute  the  actual  force  necessary  to  work  a  pump,  (ex- 
clusive of  the  pump  rods,)  let  the  height  of  a  discharging  spout 
S,  above  the  level  of  the  water  in  the  well,  be  expressed  in  feet, 
and  let  the  number  which  expresses  it  be  h.  Let  the  diameter 
of  the  piston,  expressed  in  parts  of  a  foot,  be  d ;  then  the  section 
of  the  piston  expressed  in  parts  of  a  square  foot,  will  be  d2x.7854. 
If  this  product  be  multiplied  by  the  number  of  feet  h  in  the 
height,  we  shall  obtain  the  number  of  cubic  feet  of  water 
which  it  is  necessary  to  lift  at  each  stroke,  since  this  number= 
rfx.7854x/t.  Now  each  cubic  foot  of  water  weighs  about  62£ 
pounds;  hence  d?>  7854x7ix62s=:  number  of  pounds  required 
at  each  stroke  to  lifl  the  piston. 

The  column  of  water  discharged  at  each  stroke,  is  equal  to  a 
column  of  water,  whose  base  is  the  section  of  the  piston,  and 
whose  altitude  is  the  length  of  the  stroke.  The  quantity  may 
therefore  be  found,  in  cubic  feet, by  multiplying  t?x.7854  by  the 
number  of  feet  in  the  length  of  the  stroke.  The  weight  of  the 
water  discharged  may  be  ascertained  in  pounds  avoirdupois,  by 
multiplying  this  product  by  62£. 

497.  THE  FORCING  PUMP.— A    cylinder  ABC,   (Fig.   205,)    is 
placed  with  its  lower  end  C  in  the  reservoir.     It  has  a  fixed  valve 
at  V,  opening  upward,  and  a  solid  piston  without  a  valve,  play- 

*  Library  of  Useful  Knowledge,  Art. '  Pneumatics.' 


PNEUMATICS. 


ing  air-tight  in  the  upper  barrel  AB.  It  is  con- 
nected with  another  barrel  DE  by  a  valve  V 
opening  upward  and  outward.  The  tube  DE 
is  carried  to  whatever  height  it  may  be  necessary 
to  elevate  the  water.  Let  us  suppose  that  the 
solid  piston  P  is  in  contact  with  the  valve  V,  and 
that  the  water  in  the  lower  barrel  is  at  the  same 
level  C  with  the  water  in  the  reservoir.  Upon 
raising  the  piston,  the  air  in  BC  will  be  rarefied,  A 
and  the  water  will  ascend  in  BC  exactly  as  in  the 
suction  pump.  Upon  again  depressing  the  piston, 


Fig.  205. 
E 


M 


N 


D 


pis 

the  air  in  PV  will  be  depressed,  and  it  will  force 
open  the  valve  V,  and  escape  through  it.  The 
process,  therefore,  until  water  is  raised  through  V 
into  the  upper  barrel,  is  precisely  the  same  as  for 
the  suction  pump,  the  valve  V  taking  the  place 
of  the  piston- valve  in  that  machine.  Now,  let  us 
suppose  that  water  has  been  elevated  through  V, 
and  that  the  space  PV  is  filled  with  it.  Upon 
depressing  the  piston,  this  water,  not  being  per- 
mitted to  return  through  V,  is  forced  through  V, 
and  ascends  in  the  tube  DE.  By  continuing  the 
process,  water  will  accumulate  in  the  tube  DE,  until  it  acquires 
the  necessary  elevation,  and  is  discharged.  Or,  to  enunciate 
the  principle  of  this  machine  in  general  terms : 

In  the  forcing  pump,  the  piston  has  no  valve,  but  the  water  being 
elevated  into  the  exhausting  tube,  as  in  the  suction  pump,  it  is  then 
forced,  by  the  descent  of  the  piston,  into  the  ascending  pipe  through 
a  valve  placed  in  the  side  and  at  the  bottom  of  the  exhausting  tube. 

498.  The  force  requisite  to  elevate  the  piston  in  this  pump 
until  the  water  reaches  it,  is  computed  in  exactly  the  same 
manner  as  for  the  suction  pump,  and  exclusive  of  the  weight  of 
the  piston  and  its  rods,  and  the  effects  of  friction,  it  is  equal  to 
the  weight  of  a  column  of  water  whose  base  is  the  section  of  the 
piston,  and  whose  height  is  the  distance  of  the  level  of  the  water 
in  the  barrel  AC,  above  the  level  in  the  reservoir.  It  is  evident 
also  from  wrhat  has  been  said  on  the  suction  pump,  that  the  valve 
V  should  be  less  than  thirty-four  feet  above  the  level  of  the  water 
in  the  reservoir.  If  P  express  in  pounds  av.  the  weight  of  the 
piston  and  its  rods,  d  be  the  diameter  of  a  section  of  the  piston 
expressed  in  parts  of  a  foot,  and  h  be  the  number  of  feet  in  AC, 
the  force  in  pounds  necessary  to  lift  the  piston  will  be  hxd?x 
.7854X62.5+P. 

Let  us  now  examine  the  force  necessary  to  depress  the  piston. 
Let  the  level  of  the  water  in  ED  be  M.  The  atmospheric  pres- 
sure on  M  will  be  balanced  by  the  same  pressure  on  the  piston, 
by  the  power  of  transmitting  pressure  peculiar  to  fluids.  This 


340  NATURAL   PHILOSOPHY. 

force  may  therefore  be  neglected  ;  also  the  part  PV  will  balance 
the  part  ND  of  the  ascending  column.  Hence  it  appears,  that 
the  pressure  exerted  by  the  water  in  PV  on  the  lower  surface  of 
the  piston,  is  equal  to  the  weight  of  a  column  of  water  whose 
base  is  equal  to  the  section  of  the  piston,  and  whose  height  is 
MN.  This,  therefore,  is  the  force  to  be  overcome  in  the  descent 
of  the  piston,  and  the  weight  P  of  the  piston  and  its  rods  assists  in 
overcoming  it.  Let  h'  be  the  number  of  feet  in  MN,  and  the 
mechanical  force  necessary  to  be  applied  to  depress  the  piston 
will  be  expressed  in  pounds  by  h'  x^x. 7854x62.5— P. 

From  these  observations,  it  appears  that  the  weight  of  the  piston 
and  its  rods  assists  the  forcing  power  of  the  machine,  but  opposes 
its  suction  power.  These  effects,  therefore,  on  the  whole,  neu- 
tralize one  another.* 

499.  The  entire  force  used  in  raising  the  water,  will  be  found 
by  adding  the  force  necessary  to  elevate  the  piston  to  that  which 
is  necessary  to  depress  it.     As  in  this  case  the  weight  of  the 
piston  and  rods  increases  the  one  as  much  as  it  diminishes  the 
other,  the  entire  force  will  be  the  weight  of  a  column  of  water 
whose  base  is  the  section  of  the  piston,  and  whose  height  is  PC 
-f  MN,  that  is,  the  height  of  the  level  of  the  water  in  tjie  ascend- 
ing pipe  above  the  level  of  the  water  in  the  reservoir ;  and  ex- 
pressed in  pounds,  this  is  (h-\-h')  x. 7854x^x62.5. 

It  appears,  therefore,  that,  other  circumstances  being  the  same, 
the  power  of  the  forcing  pump  has  the  advantage  over  that  of  the 
suction  pump,  by  the  weight  of  the  piston  and  its  rods. 

500.  In  forcing  pumps,  since  the  power  is  applied  by  separate 
impulses,  the  water  would  issue  in  jets  were  not  some  contrivance 
adopted  to  equalize  its  flow  from  the  tube.     This  purpose  is  ef- 
fected by  means  of  an  air-vessel,  in  which  a  portion  of  condensed 
air  is  made  the  medium  of  communication.     The  force  imparted 
by  successive  blows  of  the  piston  is  first  received  by  this  con- 
fined body  of  air,  and  this,  by  its  elasticity,  reacts  on  the  surface 
of  the  water  in  the  air-vessel,  and  forces  it  out  by  the  conducting 
pipe  or  hose. 

An  example  of  this  is  afforded  in  the  Fire  Engine.  The  fire 
engine  consists  of  two  forcing  pumps,  which  throw  the  water  into 
an  air-vessel,  from  which  it  is  thrown  out  of  the  conducting  hose 
by  the  elastic  pressure  of  condensed  air.  Thus,  (Fig.  206,)  AB, 
AB  are  two  forcing  pumps,  whose  pistons  P,  P,  are  wrought  by 
a  beam  whose  fulcrum  is  at  F ;  V,  V,  are  valves  which  open 
upward  from  a  suction  tube  T  which  communicates  with  a 
reservoir ;  t,  t,  are  force  pipes,  which  communicate  by  valves 
V',  V,  opening  into  an  air  vessel  M.  A  tube  L  is  inserted 

*  Library  of  Useful  Knowledge. 


PNEUMATICS. 


341 


Fig.  206. 


in  the  top  of  this  vessel,  terminating 
in  a  leathern  tube  or  hose,  through 
which  the  water  is  forced  by  the 
pressure  of  air  confined  in  M,  which, 
in  consequence  of  its  elasticity,  acts 
nearly  uniformly  on  the  surface  of  the 
water,  and  forces  it  through  the  hose 
in  a  continual  stream. 

501.  THE  HUNGARIAN  MACHINE. — 
This  celebrated  machine  is  employed 
in  draining  a  mine  at  Chemnitz,  in 
Hungary.  We  introduce  a  description 
of  it  here,  on  account  of  its  affording  a 
good  illustration  of  several  hydraulic  T 
and  pneumatic  principles.  Here  the 
object  is,  to  raise  water  from  a  deep  mine  to  the  height  of 
ninety-six  feet,  where  it  can  be  poured  off  by  a  horizontal  chan- 
nel. Now  it  is  easy  in  such  a  case,  to  take  a  stream  of  water 


near  the  top  of  the  pit,  (and  a  very 
small  stream  will  answer  the  pur- 
pose,) and  to  convey  it  into  a  pipe 
which  shall  descend  into  the  mine, 
and  afford,  by  hydrostatic  pressure, 
any  degree  of  force  required  to  raise 
the  water  of  the  mine,  not  indeed  to 
the  top  of  the  pit,  for  that  is  hardly 
ever  necessary,  but  to  such  a  height 
that  it  may  be  poured  off  by  a  hori- 
zontal drain.  In  the  mine  of  Hun- 
gary, the  water,  which  is  to  supply 
the  required  pressure,  is  taken  at  the 
height  of  two  hundred  and  sixty  feet 
above  the  surface  of  the  water  in  the 
pit.  From  the  cistern  A,  where  the 
head  of  water  collects,  it  enters  the 
perpendicular  pipe  B,  which  de- 
scends nearly  to  the  bottom  of  the  air- 
vessel  C.  Flowing  into  this,  it  con- 
denses the  air  before  it,  which,  by  its 
elasticity,  receives  and  exerts  the 
whole  force  created  by  the  pressure 
of  the  column  of  water  B.  This 
force  is  transmitted  through  the  air- 
pipe  D,  to  the  surface  of  the  water 
contained  in  the  well  E,  which  is 
sunk  into  the  water  of  the  mine,  ad- 
mitting it  freely  by  means  of  a  valve 
in  the  bottom  opening  upward.  This 


Fig.  207. 


NATURAL    PHILOSOPHY. 

well  and  the  air-vessel  C,  are  made  strong  and  air-tight.  From 
near  the  bottom  of  the  well  proceeds  a  perpendicular  tube  F, 
reaching  to  the  height  of  the  drain. 

502.  We  may  now  easily  understand  the  operation  of  the  ma- 
chine.    We  have  to  raise  water  ninety-six  feet,  and  we  can  com- 
mand a  column  of  water  two  hundred  and  sixty  feet  high  ;  but 
we  have  no  occasion  to  employ  the  whole  of  this  force,  and  so 
long  a  column  of  water  would  require  a  pipe  of  very  great 
strength,  especially  in  the  lower  parts  of  it.     A  column  one  hun- 
dred 'fend  thirty-six  feet  long,  is  found  by  calculation  competent 
to  raise  the  water  in  the  pit  to  the  required  height  of  ninety-six 
feet,  and  to  make  it  flow  off  with  considerable  velocity  into  the 
drain.     At  the  distance  of  one  hundred  and  thirty-six  feet  from 
the  reservoir,  we  interpose  an  air-vessel  C,  and  receive  the  entire 
force  of  the  column  B  upon  the  air  of  this  vessel,  which  is  com- 
pressed into  a  small  space  in  the  upper  part  of  the  vessel,  and  has 
its  elasticity  proportionally  augmented.  (Art.  452'.)     This  force, 
by  means  of  the  pipe  D,  is  transmitted  to  the  surface  of  the  wa- 
ter in  the  well,  and  forces  the  water  up  the  pipe  F,  which  deliv- 
ers it  into  the  drain.     The  principle  of  the  Hungarian  machine, 
therefore,  may  be  thus  enunciated : 

Water  is  raised  by  the  pressure  of  a  column  of  water,  longer 
than  the  column  required  to  be  raised,  and  at  a  higher  level ;  the 
pressure  being  transmitted  from  one  column  to  the  other,  through 
the  medium  of  condensed  air.* 

When  the  water  contained  in  the  cistern  E  is  raised  and  de- 
livered through  the  pipe  F,  the  pressure  upon  E  is  relieved  by 
opening  the  stop-cock  n,  and  closing  /.  Water  is  again  let  in  by 
opening  the  stop-cock  Z,  and  closing  n,  and  thus  the  process  is 
continued. 

503.  STEAM-ENGINE.- -It  belongs  to  Chemistry  to  investigate 
the  properties  of  steam,  and  to  Natural  Philosophy  to  apply  it 
as  a  mechanical  agent.     The  steam-engine  is  the  fruit  of  the 
highest  efforts  of  both  these  sciences,  and  the  most  valuable  pres- 
ent ever  made  by  philosophy  to  the  arts.f     As  it  is  impossible 
clearly  to  understand  the  principles  and  construction  of  this  en- 
gine, without  a  knowledge  of  the  properties  of  steam,  on  which 
they  depend,  we  subjoin  an  account  of  a  few  of  its  leading  prop- 

*  A  remarkable  fact  is  mentioned  in  connection  with  the  Hungarian  Machine,  which 
shows  very  strikingly  the  increase  of  capacity  for  heat  and  consequent  production  of 
cold,  which  arises  from  a  sudden  enlargement  of  volume.  When  the  efflux  of  the 
water  from  the  pipe  F  has  ceased,  if  the  cock  of  the  air-vessel  C  be  opened,  the  wa- 
ter and  air  rush  out  together  with  prodigious  violence,  and  the  drops  of  water  are 
changed  into  hail  or  lumps  of  ice.  It  is  a  sight  usually  shown  to  stranger^,  who  are 
desired  to  hold  their  hats  to  receive  the  blasts  of  air ;  the  ice  comes  out  with  such 
violence  as  frequently  to  pierce  the  hat  like  a  bullet. — Gregory's  Mechanics,  II,  221. 

t  Dr.  Black. 


PNEUMATICS.  343 

erties,  referring  to  chemical  authors*  for  a  more  detailed  view 
of  the  subject. 

(l.)f  The  great  and  peculiar  property  of  steam,  on  which  its 
mechanical  agencies  depend,  is  its  power  of  creating  at  one  mo- 
ment a  high  degree  of  elastic  force,  and  losing  it  instantaneously 
the  next  moment.  This  force,  acting  on  the  bottom  of  the  piston 
which  moves  in  the  main  cylinder,  raises  it,  and  fills  the  space 
below  it  with  steam.  The  steam  is  suddenly  condensed,  and 
hence  no  obstacle  is  opposed  to  the  descent  of  the  piston,  but  it 
is  readily  forced  down  again  by  steam  acting  from  above.  This 
alternate  motion  of  the  piston,  the  rod  of  which  is  connected 
with  the  working  beam,  is  all  that  is  required  in  order  to  com- 
municate motion  to  all  parts  of  the  engine. 

(2.)  The  elastic  force  of  steam  depends  on  its  temperature  and 
density  conjointly ;  and  the  temperature  necessary  to  its  production 
depends  upon  the  pressure  incumbent  upon  the  water  during  its  for- 
mation. 

The  reason  why  water  boils  at  the  temperature  of  212°  is, 
that  at  that  temperature  the  vapor  acquires  just  elasticity  suffi- 
cient to  overcome  the  atmospheric  pressure.  Hence,  steam  pro- 
duced at  the  temperature  of  boiling  water,  has  a  force  equal  to 
the  pressure  of  the  atmosphere.  When  formed  at  a  lower  tem- 
perature, its  elasticity  diminishes  in  a  geometrical  ratio,  and  in- 
creases in  the  same  ratio  when  it  is  formed  at  a  higher  tempera- 
ture. Water  boils,  or  is  converted  into  vapor,  at  a  temperature 
less  than  212°.  on  high  mountains,  (Art.  457,)  or  under  the  re- 
ceiver of  an  air-pump,  or  in  other  situations  where  the  pressure 
of  the  atmosphere  is  diminished ;  and  in  a  vacuum  the  boiling 
point  of  water  is  as  low  as  72°. 

(3.)  Heat  rapidly  augments  the  elasticity  of  steam  by  increasing 
its  density.  If  we  introduce  a  few  grains  of  water  into  a  flask, 
and  place  it  over  the  fire,  the  water  will  soon  be  converted  into 
steam,  which  will  expel  the  air  of  the  vessel  and  fill  its  whole 
capacity.  If  we  now  close  the  orifice  of  the  flask  and  continue 
the  heat,  the  steam  will  increase  in  elastic  force  in  the  same 
manner  as  air  would  do  under  similar  circumstances,  which  is 
at  a  comparatively  moderate  rate,  so  that  it  might  be  heated  red 
hot  without  exerting  any  very  violent  force.  If,  however,  the 
vessel  is  partly  filled  with  water,  and  the  heat  is  continued  as 
before,  then  the  elastic  force  is  rapidly  augmented,  and  becomes 
at  length  so  great  as  to  burst  almost  any  vessel  that  can  be  pro- 
vided ;  for  every  new  portion  of  vapor  that  is  raised  from  the 
surface  of  the  water,  adds  to  the  density  of  that  which  was  be- 
fore in  the  vessel,  and  proportionally  increases  its  elasticity.  In 
the  experiments  of  Mr.  Perkins,  a  confined  portion  of  steam,  not 

*  See,  especially,  Silliman's  Chemistry,  Vol.  I. 

t  See  Review  of  Renwick  on  the  Steam- Engine.  American  Journal  of  Science 
Vol  xx,  326. 


344 


NATURAL   PHILOSOPHY. 


in  contact  with  water,  was  heated  to  the  temperature  of  1400®, 
and  still  its  pressure  did  not  exceed  that  of  five  atmospheres  ; 
but,  by  injecting  more  water,  although  the  temperature  was 
lessened,  the  elastic  force  was  gradually  increased  to  one  hun- 
dred atmospheres.* 

The  elastic  force  of  steam  at  250°  is  twice  that  at  212°,  so  that 
the  addition  of  only  38°  of  heat  doubles  the  power ;  25°  more 
trebles  it,  and  18°  more  quadruples  the  force.  Thus  steam  at 
293°,  has  four  times  the  elastic  force  of  steam  at  212°  ;  and 
steam  at  510°  would  be  50  times  stronger,  and  would  exert  a 
force  of  750  Ibs.  to  the  square  inch,  or  108,000  Ibs.f  to  the  square 
foot.  Hence,  the  elastic  force  increases  much  faster  than  the 
temperature,  and  decreases  very  fast  as  we  descend  the  scale 
below  the  boiling  point.  Vapor  is  formed  at  every  temperature, 
even  from  ice  and  snow ;  but  in  this  case  it  differs  from  that 
formed  at  the  higher  temperatures  in  possessing  but  very  little 
elasticity,  as  will  be  apparent  from  the  following  table  : 
Table  of  the  elastic  force  of  Vapor  from  32°  to  80°. 


Tempe- 
rature. 

Force  of  vapor  in 
inches  of  mercury. 

Tempe- 
rature. 

Force  of  vapor  in 
inches  of  mercury. 

32° 

0.2000 

57° 

0.4657 

33 

0.2066 

58 

0.4832 

34 

0.2134 

59 

0.5012 

35 

0.2204 

60 

0.5200 

36 

0.2277 

61 

0.5377 

37 

0.2352 

62 

0.5560 

38 

0.2429 

63 

0.5749 

39 

0.2509 

64 

0.5944 

40 

0.2600 

65 

0.6146 

41 

0.2686 

66 

0.6355 

42 

0.2775 

67 

0.6571 

43 

0.2866 

68 

0.6794 

44 

0.2961 

69 

0.7025 

45 

0.3059 

70 

0.7260 

46 

0.3160 

71 

0.7507 

47 

0.3264 

72 

0.7762 

48 

0.3372 

73 

0.8026 

49 

0.3483 

74 

0.8299 

50 

0.3600 

75 

0.8581 

51 

0.3735 

76 

0.8873 

52 

0.3875 

77 

0.9175 

53 

0.4020 

78 

0.9487 

54 

0.4171 

79 

0.9809 

55 

0.4327 

80 

1.0120 

56 

0.4489 

*  Renwick  on  the  Steam-Engine,  p.  95. 

i  Webster's  Prisi:iples  of  Hydrostatics,  p.  186 


PNEUMATICS.  345 

(4.)  The  space  into  which  a  given  quantity  of  water  is  expanded 
in  becoming  steam,  depends  upon  the  temperature,  and  of  course 
upon  the  degree  of  pressure,  at  which  it  is  formed.  Water  con- 
verted into  steam  at  the  temperature  of  212°,  expands  nearly 
one  thousand  and  seven  hundred*  times  ;  but  when  so  confined 
as  to  be  heated  to  419cVits  volume  is  only  thirty-seven  times  that 
of  the  water  from  which  it  is  formed.  According  to  Dr.  Thom- 
son,! at  a  temperature  not  much  higher  than  500°,  steam  would 
not  much  exceed  double  the  bulk  of  the  water  from  which  it  is 
generated.  The  expansive  force  of  such  steam  would  be  truly 
formidable.  It  would,  when  it  issued  into  the  atmosphere,  sud- 
denly expand  six  hundred  and  fifty  times.  We  do  not  know  at 
what  temperature  water  would  become  vapor  without  any  in- 
crease of  volume,  but  we  can  estimate  that  it  would  then  sup- 
port a  column  of  mercury  three  thousand  two  hundred  and  forty- 
three  feet  (or  mor*  than  half  a  mile)  high,  and  would  exert  a 
pressure  of  nearly  twenty  thousand  pounds  on  every  square  inch. 

(5.)  The  absolute  quantity  of  heat  is  always  the  same  in  the  same 
weight  of  steam,  whatever  may  be  its  temperature.  When  vapor  is 
formed  at  a  low  temperature,  nearly  all  the  heat  that  enters  it  is 
in  the  latent  state  ;  but  as  we  heat  it  to  a  higher  degree,  its  pro- 
portion of  sensible  heat  is  constantly  augmented,  and  that  of 
latent  heat  diminished  in  the  same  ratio,  so  that  the  sum  of  the 
two  is  the  same  constant  quantity. 

These  preliminary  principles  being  well  understood  and  kept 
clearly  in  mind,  it  will  be  easy  for  the  learner  to  comprehend  the 
principles  involved  in  the  steam-engine,  and  the  dangers  with 
which  it  is  environed.  The  general  interest  felt  in  this  subject 
renders  it  one  peculiarly  deserving  of  the  attention  of  the  stu- 
dent, and  induces  us  to  devote  a  considerable  space  to  the  con- 
sideration of  it. 

504.  The  steam-engine  owes  its  present  form  and  perfection, 
chiefly  to  the  genius  and  labors  of  the  late  James  Watt,  Esq.,  of 
England.  His  inquiries  on  the  subject  commenced  in  the  year 
1763.  The  engine  in  use  previous  to  that  time,  was  what  is  now 
called  the  Atmospheric  Engine.  It  has  already  been  remarked, 
(Art.  503,)  that  the  chief  object  in  the  use  of  steam  is  to  cause 
the  alternate  ascent  and  descent  of  a  piston  moving  in  a  cylin- 
der, since  this  motion  may,  by  the  aid  of  machinery,  be  so  modi- 
fied as  to  answer  all  the  purposes  required  of  the  engine.  In 
the  atmospheric  engine,  at  the  commencement  of  the  operation, 
the  piston  remained  drawn  up  to  the  top  of  the  cylinder,  being 
kept  there  by  the  preponderance  of  the  opposite  arm  of  the  lever, 
or  working  beam,  to  which  it  was  attached.  Steam  being  ad- 

*  It  will  assist  the  memory  to  consider  a  cubic  inch  of  water  as  forming  a  cubic 
foot  of  steam,  as  is  nearly  the  fact. 

t  Outlines  of  the  Sciences  of  Heat  and  Electricity,  p.  22S 

44 


846  NATURAL   PHILOSOPHY. 

mitted  through  a  valve  into  the  cylinder,  expelled  the  air  and 
occupied  its  place.  Cold  water  now  being  admitted,  the  steam 
was  suddenly  condensed,  a  vacuum  formed,  and  the  atmospheric 
pressure  on  the  upper  side  of  the  piston,  having  nothing  to  coun- 
terbalance it  on  the  lower  side,  forced  it  down  to  the  bottom  of 
the  cylinder.  Steam  being  again  admitted  below  the  piston, 
supplied  an  upward  force  equivalent  to  the  downward  pressure 
of  the  atmosphere  on  the  piston,  and  the  preponderance  of  the 
opposite  arm  of  the  lever  dragged  up  the  piston  as  before. 

It  is  impossible  to  understand  the  reason  of  the  construction 
of  the  different  parts  of  Watt's  steam-engine,  without  a  know- 
ledge of  the  imperfections  of  the  atmospheric  engine, — imper- 
fections for  which  he  sought  and  found  a  complete  remedy.  We 
therefore  subjoin  a  brief  notice  of  the  successive  steps  by  which 
Mr.  Watt  was  led  to  his  great  improvements.* 

505.  Previous  to  the  discoveries  of  Watt,  the  energy  of  this 
power  was  well  known  ;  but  the  waste  of  fuel  consumed  in 
generating  it  was  so  enormous,  as  to  render  it,  in  most  situations, 
a  more  expensive  power  than  the  other  forces  generally  employed 
to  jmove  machinery.  The  reason  of  this  waste  will  be  easily 
understood.  When  the  steam  fills  the  cylinder,  so  as  to  balance 
the  atmospheric  pressure  on  the  piston,  the  cylinder  must  have 
the  same  temperature  as  the  steam  itself.  Now,  on  introducing 
the  condensing  jet,  the  steam  mixed  with  this  water,  forms  a 
mass  of  hot  water  in  the  bottom  of  the  cylinder.  This  water, 
not  being  under  the  atmospheric  pressure,  boils  at  a  very  low 
temperature,  (Art.  457,)  and  produces  a  vapor  which  resists  the 
descent  of  the  piston.  The  heat  of  the  cylinder  itself  assists  this 
process  ;  so  that  in  order  to  produce  a  tolerably  perfect  vacuum 
under  the  piston,  it  was  found  necessary  to  introduce  so  consid- 
erable a  quantity  of  condensing  water,  as  would  reduce  the  tem- 
perature of  the  water  in  the  cylinder  lower  than  100°,  and  which 
would  consequently  cool  the  cylinder  itself  to  that  temperature. 
Under  these  circumstances,  the  descent  of  the  piston  was  found 
to  suffer  very  little  resistance  from  any  vapor  within  the  cylin- 
der. But  then  on  the  subsequent  ascent  of  the  piston,  an  im- 
mense waste  of  steam  ensued  ;  for  on  being  admitted  under  the 
piston,  the  cold  cylinder  and  water  of  condensation  immediately 
condensed  the  steam,  and  continued  to  do  so,  until  the  cylinder 
became  heated  again  up  to  212°,  to  which  point  the  whole  cyl- 
inder must  be  again  heated,  before  the  steam  would  acquire  suf- 
ficient elasticity  to  raise  the  piston.  Here  then  was  an  obvious 
and  an  extensive  source  of  the  waste .  of  heat.  At  every  de- 
scent of  the  piston,  the  cylinder  must  be  cooled  to  below  100°  ; 
and  at  every  ascent,  it  must  be  again  heated  to  212°.  It  there- 

*  Abridged  from  Lardner's  Lectures  on  the  Steam-Engine. 


PNEUMATICS.  347 

fore  became  a  question,  whether  the  force  gained  by  the  in- 
creased perfection  of  the  vacuum  was  adequate  to  the  waste  of 
fuel  in  producing  the  vacuum  ;  and  it  was  found,  on  the  whole, 
more  profitable  not  to  cool  the  cylinder  to  so  low  a  temperature, 
and  consequently  to  work  with  a  very  imperfect  vacuum,  and  a 
diminished  power.  Watt,  therefore,  found  the  atmospheric  en- 
gine in  this  dilemma,  either  much  or  little  water  of  condensation 
must  be  used.  If  much  were  used,  the  vacuum  would  be  per- 
fect ;  but  then  the  cylinder  would  be  cooled,  and  would  occasion 
an  extensive  waste  of  fuel  in  heating  it.  If  little  were  used,  a 
vapor  would  remain,  which  would  resist  the  descent  of  the  pis- 
ton, and  rob  the  atmosphere  of  part  of  its  power.  The  great 
problem  then  pressed  itself  on  his  attention,  to  condense  the  steam 
without  cooling  the  cylinder.  To  the  solution  of  this  problem, 
Watt  now  gave  his  whole  mind.  The  idea  occurred  to  him  of 
providing  a  vessel  separate  from  the  cylinder,  in  which  a  con- 
stant vacuum  might  be  kept  up.  If  a  communication  could  be 
opened  between  the  cylinder  and  this  vessel,  the  steam,  by  its 
expansive  property,  would  rush  from  the  cylinder  to  this  vessel, 
where,  being  exposed  to  cold,  it  would  be  immediately  con- 
densed, the  cylinder  meanwhile  being  sustained  at  the  tempera- 
ture of  212°.  This  happy  conception  formed  the  first  step  of 
that  brilliant  career,  which  has  immortalized  the  name  of  Watt, 
and  spread  his  fame  throughout  the  civilized  world.  He  states 
that  the  moment  the  notion  of  "  separate  condensation"  struck 
him,  all  the  other  details  of  his  improved  engine  followed  in 
rapid  and  immediate  succession  ;  so  that  in  the  course  of  a  day, 
his  invention  was  so  complete,  that  he  proceeded  to  submit  it  to 
experiment. 

506.  His  first  notion  was,  as  has  been  stated,  to  provide  a  sep- 
arate vessel  called  a  condenser,  having  a  pipe  or  tube  communi- 
cating with  the  cylinder.  This  condenser  he  proposed  to  keep 
cold  by  immersing  it  in  a  cistern  of  cold  water,  and  by  providing 
a  jet  of  cold  water  to  play  within  it.  When  the  communication 
with  the  cylinder  is  opened,  the  steam,  rushing  into  the  con- 
denser, is  immediately  condensed  by  the  jet  and  the  cold  surface. 
But  here  a  difficulty  presented  itself,  viz.  how  to  dispose  of  the 
condensing  water  and  condensed  steam,  which  would  collect  in 
the  bottom  of  the  condenser.  Besides  this,  a  certain  quantity 
of  air  would  inevitably  enter,  mixed  with  the  condensing  water, 
which,  accumulating,  would  collect  in  the  cylinder,  and  resist 
the  descent  of  the  piston.  To  remedy  this,  he  proposed  to  form 
a  communication  between  the  bottom  of  the  condenser  and  a  pump, 
which  he  called  the  AIR-PUMP  ;  so  that  the  water,  the  air,  and  the 
other  fluids,  which  might  be  collected  in  the  condenser,  would  thus 
be  drawn  off  ;  and  this  pump  could  be  worked  by  the  machine  itself. 


348  NATURAL   PHILOSOPHY. 

507.  Another  inconvenience  was  still  to  fee  removed.     On  the 
descent  of  the  piston,  the  air  which  entered  the  cylinder  from 
above  would  lower  its  temperature  ;  so  that,  upon  the  next  as- 
cent, some  of  the  steam  which  entered  it  would  be  condensed, 
and  hence  would  arise  a  source  of  waste.     To  remove  this  diffi- 
culty, Watt  proposed  to  close  the  top  of  the  cylinder  altogether, 
by  an  air-light  and  steam-tight  cover,  allowing  the  piston  rod  to 
play  through  a  hole  furnished  with  a  stuffing  box,  and  to  press 
down  the  piston  by  steam  instead  of  the  atmosphere.     Watt's  grand 
improvements  in  the  steam-engine  consisted,  therefore,  of  three 
separate  steps.     The  first  was  the  introduction  of  the  condenser  ; 
the  second,  the  contrivance  of  the  air-pump ;  and  the  third,  the 
employment  of  steam  instead  of  atmospheric  pressure,  to  force 
down  the  piston.     This  third  step  totally  changed  the  character 
of  the  machine.     It  now  became  really  a  steam-engine  in  every 
sense  ;  for  the  pressure  above  the  piston  was  the  elastic  force 
of  steam,  and  the  vacuum  below  it  was  produced  by  the  con- 
densation of  steam ;  so  that  steam  was  used  both  directly  and 
indirectly  as  a  moving  power  ;  whereas,  in  the  atmospheric  en- 
gine, the  indirect  force  of  steam  only  was  used,  being  adopted 
merely  as  an  easy  method  of  producing  a  vacuum. 

508.  The  last  difficulty  respecting  the  economy  of  heat  that 
remained  to  be  removed,  arose  from  the  liability  of  the  external 
surface  of  the  cylinder  to  become  cool  by  the  circulation  of  the 
cold  air  around  it.     To  obviate  this  difficulty,  Mr.  Watt  first 
proposed  casing  the  cylinder  in  wood,  as  being  a  substance  which 
conducts  heat  slowly.     He  subsequently,  however,  adopted  a 
different  method,  and  enclosed  one  cylinder  within  another,  leav- 
ing a  space  between  them,  which  he  kept  constantly  supplied 
with  steam.     Thus  the  inner  cylinder  was  kept  constantly  up  to 
the  temperature  of  the  steam  which  surrounded  it.     The  outer 
cylinder  is  called  the  jacket. 

Watt  computed  that  in  the  atmospheric  engine,  three  times  as 
much  heat  was  wasted  in  heating  the  cylinder,  and  the  other 
parts  of  the  machine,  as  was  spent  in  useful  effect.  And  since, 
in  the  improvements  proposed  by  him,  nearly  all  the  waste  was 
removed,  he  contemplated,  and  afterward  actually  effected,  a 
saving  of  three  fourths  of  the  fuel. 

With  these  things  distinctly  in  view,  the  learner  will  now  be 
prepared  to.  understand  the  construction  of  this  noble  engine,  in 
its  most  improved  state. 

509.  The  difficulty  of  understanding  the  construction  and  prin- 
ciples of  the  steam-engine,  (as  is  the  case  also  with  many  other 
machines  where  the  parts  are  numerous,)  is  greatly  enhanced  by 
the  variety  of  accidental  trappings  or  appendages  that  are  em- 
ployed about  the  machine,  to  perform  subordinate  offices.     As 


PXECMATICS. 


349 


these  render  the  comprehension  of  the  leading  principles  difficult, 
when  the  explanation  is  attempted  from  the  engine  itself,  so  these 
inferior  parts  are  often  so  multiplied  in  diagrams  as  greatly  to 
obscure  the  representation.  We  shall  begin  our  explanation 
with  a  diagram  which  presents  the  naked  principles,  divested  of 
all  unnecessary  appendages. 

510.  The  chief  parts  of  the  engine  are  the  boiler  A,  the  cylin- 
der C,  the  condenser  L,  and  the  air-pump  M.  B  is  the  steam  pipe, 
branching  into  two  arms,  communicating  respectively  with  the 
top  and  bottom  of  the  cylinder ;  anc  K  is  the  eduction  pipe, 
formed  of  the  two  branches  which  proceed  from  the  top  and 
bottom  of  the  cylinder,  and  communicate  between  the  cylinder 
and  the  condenser.  N  is  a  cistern  or  well  of  cold  water,  in 
which  the  condenser  is  immersed.  Each  branch  of  pipe  has  its 
own  valve,  as  F,  G,  P,  Q,  which  may  be  opened  or  closed  as  the 
occasion  requires. 

Fig.  208* 


511.  Suppose,  first,  that  all  the  valves  are  open,  while  steam  is 
issuing  freely  from  the  boiler.  It  is  easy  to  see  that  the  steam 
would  circulate  freely  through  all  parts  of  the  machine,  expelling 
the  air,  which  would  escape  through  the  valve  in  the  piston  of 
the  air-pump,  and  thus  the  interior  spaces  would  be  all  filled  with 
steam.  This  process  is  called  blowing  through  ;  it  is  heard  when 
a  steamboat  is  about  setting  off.  Next  the  valves  F  and  Q  are 
closed,  G  and  P  remaining  open.  The  steam  now  pressing  on 
the  cylinder,  forces  it  down,  and  the  instant  when  it  begins  to  de- 
scend, the  stop-cock  O  is  opened,  admitting  cold  water,  which 
meets  the  steam  as  it  rushes  from  the  cylinder  and  effectually  con- 
denses it,  leaving  no  force  below  the  piston  to  oppose  its  descent. 


*  From  Jones's  Conversations  on  Chemistry,  a  work  which  contains  a  very  lumin- 
ous view  of  the  elementary  principles  of  the  steam-engine. 


350  NATURAL   PHILOSOPHY. 

Lastly,  G  and  P  being  closed,  F  and  Q  are  opened,  the  steam 
flows  in  below  the  piston  and  rushes  from  above  it  into  the  con- 
denser, by  which  means  the  piston  is  forced  up  again  with  the 
same  power  as  that  with  which  it  descended.  Meanwhile  the 
air-pump  is  playing,  and  removing  the  water  and  air  from  the 
condenser,  and  pouring  the  water  into  a  reservoir,  whence  it  is 
conveyed  to  the  boiler  to  renew  the  same  circuit. 

512.  The  kind  of  valve  chiefly  employed  in  the  steam-engine 
is  that  called  the  puppet  valve.     It  resembles  the  stopper  of  a 
decanter,  but  is  more  obtuse.     All  these  various  appendages  of 
the  machine,  are  carried  by  the  engine  itself;  the  air-pump  is 
worked  by  having  its  piston  rod  attached  to  one  arm  of  the 
working  beam,  and  the  valves  are  opened  at  the  instant  required 
by  means  of  levers,  to  which  also  motion  is  communicated  from 
the  same  source. 

513.  Soon  after  the  invention  of  these  engines,  Watt  found 
that,  in  some  instances,  inconvenience  arose  from  the  too  rapid 
motion  of  the  steam  piston  at  the  end  of  its  stroke,  owing  to  its 
being  moved  with  an  accelerated  motion.*     This  was  owing  to 
the  uniform  action  of  the  steam  pressure  upon  it.     For  on  first 
putting  it  in  motion  at  the  top  of  the  cylinder,  the  motion  was 
comparatively  slow,  but  from  the  continuance  of  the  same  pres- 
sure, the  velocity  with  which  the  piston  descended  was  continu- 
ally increasing,  until  it  reached  the  bottom  of  the  cylinder,  when 
it  acquired  its  greatest  velocity.     To  prevent  this,  and  to  render 
the  descent  as  nearly  uniform  as  possible,  it  was  proposed  to  cut 
oft'  the  steam  before  the  descent  was  completed,  so  that  the  re- 
mainder might  be  effected  merely  by  the  expansion  of  the  steam 
which  was  admitted  to  the  cylinder.     To  accomplish  this,  he 
contrived  by  means  of  a  pin  on  the  rod  of  the  air-pump,  to  close 
the  upper  steam  valve  when  the  steam  piston  had  completed  one 
third  of  its  entire  descent,  and  to  keep  it  closed  during  the  remain- 
der of  that  descent,  and  until  the  piston  again  reached  the  top  of 
the  cylinder.     By  this  arrangement,  the  steam  pressed  the  piston 
with  its  full  force  through  one  third  of  the  descent,  and  thus  put 
it  into  motion  ;  during  the  other  two  thirds  of  the  way,  the  steam 
thus  admitted  acted  merely  by  its  expansive  force,  which  became 
less  in  exactly  the  same  proportion  as  the  space,  given  to  it  by  the 
descent  of  the  piston,  increased.     Thus,  during  the  last  two  thirds 
of  the  descent,  the  piston  is  urged  by  a  gradually  decreasing 
force,  which  in  practice  is  found  just  sufficient  to  keep  up  in  the 
piston  a  uniform  velocity.     Another  advantage  gained  by  this 
contrivance,  independently  of  the  uniformity  of  motion,  was,  that 

*  For  since  the  steam  continues  to  act  upon  the  piston  during  its  descent,  its  velo- 
city would  be  constantly  increased,  like  that  of  a  ball  in  the  barrel  of  a  gun. 


PNEUMATICS. 


351 


two  thirds  of  the  fuel  was  saved  ;  for  instead  of  consuming  a  cyl- 
inder full  of  steam  at  each  descent  of  the  piston,  only  one  third  of 
a  cylinder  full  was  necessary.  Steam-engines  constructed  on  this 
principle  are  said  to  act  expansively. 

514.  From  the  foregoing  account  of  the  principles  of  the  steam- 
engine,  the  learner  will  be  able  to  give  a  full  explanation  of  the 
construction  and  use  of  the  various  parts  of  this  important  and 
interesting  machine,  from  the  figure.* 

Fig.  209. 


A.  The  BOILER. 

B.  The  STEAM  PIPE,  conveying   the  steam   to    the  cylinder, 

having  a  steam-cock  b  to  admit  or  exclude  the  steam  at 
pleasure. 

C.  The  CYLINDER,  surrounded  with  the  jacket,  cc. 

D.  The  EDUCTION  PIPE,  communicating  between  the  cylinder 

and  the  condenser. 

E.  The  CONDENSER,  with  a  valve  e,  called  the  Injection-cock, 

admitting  a  jet  of  cold  water,  which  meets  the  steam  the 
instant  the  latter  enters  the  condenser. 

F.  The  AIR-PUMP. 

*  One  of  the  best  descriptions  of  the  steam-engine  may  be  found  Ji  Millicgtonli 
Epitome  of  Natural  Philosophy, 


352  NATURAL   PHILOSOPHY. 

G.  G.  COLD  WATER  CISTERN,  for  the  Condenser,  filled  by 
H.  The  COLD  WATER  PUMP. 

I.  The  HOT  WELL,  containing  water  from  the  Condenser. 
K.  The  HOT  WATER  PUMP,  which  returns  the  water  of  con- 
densation to  the  boiler. 

L.  L.  LEVERS,  which  open  and  shut  the  valves  in  the  channel  be- 
tween the  Induction  Pipe,  Cylinder,  Eduction  Pipe,  and 
Condenser ;  which  levers  are  raised  or  depressed  by  pro- 
jections attached  to  the  piston  rod  of  the  Air  Pump. 
M.M.  Apparatus  for  PARALLEL  MOTION,  a  beautiful  contrivance, 
by  which  the  piston  rod  is  maintained  constantly  in  a  per- 
pendicular position,  while  the  end  of  the  working  beam 
which  carries  it  moves  in  the  arc  of  a  circle.* 
N.  N.  The  WORKING  BEAM. 
O.  O.  The  GOVERNOR.     (Art.  334.) 

P.    The  CRANK.     (Art.  339.) 
Q.  Q.  The  FLY  WHEEL.     (Art.  331.) 

The  working  beam  is  here  represented  as  acting  immediately 
upon  the  fly  wheel,  from  which,  as  from  a  reservoir,  motion  may 
be  distributed  to  all  parts  of  the  engine.  (Art.  324.)  It  is  obvi- 
ous, however,  that  the  same  end  of  the  working  beam,  instead  of 
expending  its  force  upon  the  fly  wheel,  may  be  connected  directly 
with  the  piston  rod  of  a  pump  for  raising  water,  or  with  a  hori- 
zontal shaft  with 'wheels,  as  in  the  steamboat.  In  some  steam- 
boats, particularly  those  of  a  large  size,  the  fly  wheel  is  dispensed 
with,  the  inertia  of  the  boat  itself  being  sufficient^to  regulate  the 
motion.  (Art.  330.) 

515.  In  steam-engines,  of  the  foregoing  construction,  the  pres- 
sure introduced  on  one  side  of  the  piston  derives  its  efficacy,  ei- 
ther wholly  or  in  part,  from  the  vacuum  produced  by  condensa- 
tion on  the  other  side.  This  always  requires  a  condensing  appa- 
ratus, and  a  constant  and  abundant  supply  of  cold  water.  An 
engine  of  this  kind  must  therefore  necessarily  have  considerable 
dimensions  and  weight,  and  is  inapplicable  to  uses  in  which  a 
small  and  light  machine  only  is  admissible.  If  the  condensing 
apparatus  be  dispensed  with,  the  piston  will  always  be  resisted  by 
a  force  equal  to  the  atmospheric  pressure,  and  the  only  part  of  the 
steam  pressure  which  will  be  available  as  a  moving  power,  is  that 
part  by  which  it  exceeds  the  atmospheric  pressure.  Hence,  in 
engines  which  do  not  work  by  condensation,  steam  of  a  much 
higher  pressure  than  thatt)f  the  atmosphere  is  indispensably 

*  In  the  engines  constructed  recently  at  New  York,  under  the  direction  of  Mr.  R. 
L.  Stevens,  a  substitute  for  the  parallel  motion  has  been  introduced,  that  performs 
the  task  equally  well,  and  is  much  less  complex.  On  the  head  of  the  piston  rod  a 
bar  is  fixed,  at  right  angles  to  it,  and  to  the  longitudinal  section  of  the  engine.  The 
ends  of  this  bar  work  in  guides  formed  of  two  parallel  and  vertical  bars  of  iron,  by 
which  the  upper  end  of  the  piston  rod  is  constrained  to  move  in  a  straight  line. — 
(Renwick.) 


PNEUMATICS.  353 

necessary ;  and  such  engines  are  therefore  called  high  pressure 
engines.  The  steam,  when  it  has  produced  its  effect  in  raising  or 
depressing  the  piston,  escapes  into  the  atmosphere,  not  being  con- 
densed and  returned  to  the  boiler  as  in  low  pressure  or  condensing 
engines.  In  these  engines  the  whole  of  the  condensing  apparatus, 
viz.  the  cold  water  cistern,  condenser,  air-pump,  &c.  are  dispensed 
with,  and  nothing  is  retained  except  the  boiler,  cylinder,  piston, 
and  valves.  Consequently,  such  an  engine  is  small,  light,  and 
cheap.  It  is  portable  also,  and  may  be  removed,  if  necessary, 
along  with  its  load,  and  is  therefore  well  adapted  to  locomotive 
purposes.  Hence  its  use  in  small  steamboats,  and  in  locomotive 
carriages  on  railways. 

516.  Although  the  idea  of  propelling  boats  by  the  force  of 
steam  was  entertained  by  different  individuals,  in  different  coun- 
tries, long  before  it  was  carried  into  practice,  yet  the  first  suc- 
cessful effort  at  steam  navigation  was  made  by  our  countryman, 
Robert  Fulton,  in  the  year  1807.  This  year,  the  first  steamboat, 
the  Clermont,  ascended  the  river  Hudson  to  Albany.  Fulton 
never  contemplated  a  velocity  for  steamboats  greater  than  eight 
or  nine  miles  per  hour ;  but  the  average  speed  now  given  to 
boats  on  the  Hudson  is  no  less  than  fifteen  miles  per  hour,  and 
sixteen  miles  is  no  unusual  rate.  Crossing  the  ocean  by  steam, 
as  is  now  done  with  great  speed  and  safety,  is  well  placed 
among  the  highest  enterprises  of  the  present  age.  The  first 
successful  voyages  across  the  Atlantic  by  steam,  were  performed 
by  two  steamships,  the  Great  Western  and  the  Sirius,  in  1838. 

The  prevalence  of  westerly  winds  in  the  Atlantic,  particular- 
ly at  certain  seasons  of  the  year,  rendered  the  passage  from  Eng- 
land to  the  United  States  uncertain,  and  frequently  long  and  te- 
dious ;  but  the  resistless  force  of  steam  has  triumphed  over  this 
difficulty,  and  has  made  the  passage  between  the  two  countries 
sure  and  speedy. 

Steam  is  the  most  manageable  of  all  the  forces  intrusted  to 
man.  It  is  his  sole  prerogative  to  develop  and  direct  this  pow- 
er ;  and  so  pliant  is  it,  that  it  is  ever  ready  to  perform  for  him 
the  humblest  or  the  mightiest  of  his  works. 

45 


354  NATURAL   PHILOSOPHY. 


PART  V. ACOUSTICS. 


517.  ACOUSTICS*  is  the  science  which  treats  of  the  nature  and 
laws  of  SOUND. 

It  has  for  its  object  to  explain  the  origin  or  production  of 
sounds — its  propagation  through  different  media — its  reflexion 
from  surfaces — and  the  philosophical  principles  of  music. 

CHAPTER  I. 

OF  SOUND  AND  ITS  MODES  OF  PRODUCTIONS 

518.  Vibrations,  in  the  sounding  body,  communicated  to  the  or- 
gan of  hearing,  are  the  immediate  cause  of  sound. 

Whatever  may  be  the  remoter  cause  of  sound,  vibrations  must 
be  considered  as  the  immediate  cause,  since  they  always  precede 
or  accompany  it,  and  since  whatever  affects  the  vibration  of  a 
body,  produces  a  corresponding  effect  upon  the  qualities  of  the 
sounds  which  it  emits,  while  those  bodies  whose  sounds  are  sim- 
ilar, have  something  in  common  in  their  mode  of  vibration. 

If  we  rub  our  moistened  finger  along  the  edge  of  a  drinking 
glass,  or  draw  a  bow  across  the  strings  of  a  violin,  we  can  in 
both  cases  procure  sounds  which  remain  undiminished  in  inten- 
sity, as  long  as  the  operation  by  which  they  are  excited  is  con- 
tinued. A  similar  fact  takes  place  with  respect  to  any  other 
sonorous  body,  whose  structure  is  not  destroyed  by  the  mode  of 
excitation  employed. 

519.  Though  all  bodies  may,  by  some  mode  of  excitation,  be 
made  to  sound,  there  is  a  great  difference  among  them  in  the 
intensity  of  the  sounds  which  they  produce  during  the  operation, 
and  in  the  permanence  of  these  sounds  after  the  excitation  has 
ceased.     Thus  if  we  strike  two  bells,  one  of  lead  and  the  other 
of  brass,  the  sound  of  the  lead  is  feeble  and  momentary  com- 
pared with  that  of  the  brass.     Soft  bodies,  as  wool,  cotton,  and 
down,  cannot  produce  any  sound  ;  and  those  which  are  of  the 
harder  class,  as  rocks  and  stones,  are  not  in  general  sonorous. 
Liquids,  also,  are  incapable  of  producing  much  sound.     The 
quality  of  producing  sound  belongs  to  elastic  bodies,  and  those 
are  the  most  sonorous  which  are  most  elastic.     Glass,  certain 

*  From  aitonta,  to  hear.  t  Edinburgh  Encyclopedia,  Art.  Acoustics. 


ACOUSTICS.  355 

metals,  strings  when  strained  close,  hard  wood  in  thin  layers, 
and  the  air  itself,  are  at  once  among  the  most  elastic  and  most 
sonorous  bodies.  Yet  this  quality  is  not  an  invariable  attendant 
of  elasticity.  India  rubber  is  extremely  elastic,  but  not  at  all 


520.  In  comparing  the  properties  of  these  substances,  we  shall 
find  them  distinguished  from  each  other  by  the  degree  of  vibra- 
tion which  they  are  capable  of  receiving,  and  by  the  length  of 
time  during  which  they  can  preserve  a  vibratory  motion ;  those 
substances  which  are  most  capable  of  vibration  being  most  sono- 
rous, and  those  which  can  longest  maintain  a  state  of  vibration, 
also  persevering  longest  in  emitting  sound.     Bodies,  though  of 
the  same  substance,  differ  in  these  respects  according  as  their 
form  varies  ;  those  forms  which  are  most  favorable  to  the  pro- 
duction and  continuance  of  a  vibratory  motion,  being  also  most 
favorable  to  the  production  and  permanence  of  sound.     Thus  a 
hollow  globe  of  brass  is  far  less  sonorous  than  the  hemispheres 
which  are  made  by  dividing  it  into  two  equal  parts,  since  the 
structure  of  a  globe  is  such  that  the  parts  mutually  support  each 
other,  like  a  continued  arch,  while  the  form  of  the  hemispheres, 
which  approaches  that  of  a  bell,  is  peculiarly  liable  to  a  tremu- 
lous vibratory  motion.*     Indeed,  when  a  body  sounds  power- 
fully, as  a  large  bell,  or  the  lowest  string  of  a  harpsichord,  we 
can  perceive  that  it  actually  vibrates  ;  and  even  in  cases  where 
the  vibration  is  imperceptible  to  the  naked  eye,  we  may  detect 
it  by  the  microscope,  or  by  some  other  artifice.     Thus,  if  we  put 
some  water  into  a  glass  tumbler  or  basin  and  make  it  sound,  by 
applying  the  moistened  finger  as  in  Art.  518,  the  water  will  be 
agitated.     If  we  hold  the  hand  over  the  pipe  of  an  organ,  we 
feel  a  tremulous  motion  in  the  air  passing  through  it.     Such  ex- 
periments may  be  extended  to  all  solid  bodies  by  placing  upon 
them  pieces  of  paper,  or  strewing  them  with  fine  sand. 

521.  Vibrations  recurring  at  equal  intervals,  constitute  musical 
sounds.     All  continued  sounds,  which  remain  in  any  degree  uni- 
form throughout  their  duration,  are  capable  of  being  compared 
with  each  other  in  their  degree  of  acuteness.     When  sounds  are 
equally  acute,  they  are  said  to  have  the  same  pitch ;  but  when  they 
differ  in  acuteness,  that  sound  which  is  more  shrill,  is  said  to  be 
acute,  or  to  have  a  higher  pitch  ;  and  that  which  is  less  shrill,  is 
said  to  be  grave,  and  to  have  a  lower  pitch,  or  a  deeper  tone.    A 
difference  in  pitch  forms  the  chief  character  by  which  musical 
sounds  are  distinguished  from  each  other,  and  is  the  foundation 
of  their  use  in  music.     In  unmusical  sounds,  it  generally  holds  a 
place  subordinate  to  their  other  qualities. 

•  Millington's  Natural  Philosophy,  jk  123. 


356  NATURAL   PHILOSOPHY. 

Musical  sounds  have  occupied  the  attention  of  philosophers 
more  than  any  other  class  of  sounds.  The  superior  precision 
with  which  the  ear  can  estimate  any  variation  in  pitch,  renders 
these  sounds  more  easily  compared  ;  and  the  vibrations  of  the 
sonorous  bodies  which  produce  them,  are,  on  account  of  their 
superior  simplicity  of  form,  more  easily  investigated.* 

522.  MUSICAL  STRINGS.  —  A  musical  string  is  of  a  uniform  thick- 
ness, and  is  stretched  between  two  points,  by  a  force  much  greater 
than  its  weight.     The  stretching  force  is  generally  conceived  as 
measured  by  the  weight  which  would  occasion  an  equal  tension, 
on  the  supposition  that  the  string  is  made  fast  at  one  end  and 
passes  over  a  pulley  at  the  other,  the  latter  being  loaded  with 
weights.     In  the  usual  mode  of  exciting  a  musical  string,  it  vi- 
brates on  each  side  of  its  quiescent  position,  the  extremities  being 
the  only  points  which  remain  at  rest.     The  sound  which  the 
string  gives  in  this  mode  of  vibration  is  called  its  fundamental 
sound. 

The  pitch  of  the  fundamental  sound  of  musical  strings,  is 
found  by  experience  to  depend  on  three  circumstances  ;  the 
length  of  the  string,  —  its  weight  or  quantity  of  matter,  —  and  its 
tension.  The  tone  becomes  more  acute  as  we  increase  the  ten- 
sion, or  diminish  either  the  length  or  the  weight.  The  operation 
of  these  several  circumstances  may  be  seen  in  a  common  violin. 
The  pitch  of  any  one  of  the  strings  is  raised  or  lowered  by  turn- 
ing the  screw  so  as  to  increase  or  lessen  its  tension  ;  or,  the  ten- 
sion remaining  the  same,  higher  or  lower  notes  are  produced  by 
the  same  string,  by  applying  the  fingers  in  such  a  manner  as  to 
shorten  or  lengthen  the  string  which  is  vibrating  ;  or,  both  the 
tension  and  the  length  of  the  string  remaining  the  same,  the 
pitch  is  altered  by  making  the  string  larger  or  smaller,  and  thus 
increasing  or  diminishing  its  weight. 

523.  The  time  of  a  double  vibration,  is  the  time  occupied  by 
a  string,  in  passing  from  a  point  to  which  it  is  stretched  on  one 
side  to  the  opposite  extreme,  and  returning  to  the  same  point 
again.     It  has  been  ascertained,  that  the  time  of  a  double  vibra- 
tion, expressed  in  parts  of  a  second  of  time,  will  be  found  by  the 
following  operation. 

Let  L  represent  the  Jength  of  the  string  in  inches  ;  w,  thft 
weight  of  an  inch  of  the  string  ;  t,  a  weight  equivalent  to  the 
force  of  tension  ;  g,  the  rate  of  a  falling  body  =  193  ;  and  T  the 
time  of  a  double  vibration  expressed  in  seconds.  Then 


As  the  distance  of  the  string  from  its  quiescent  position  doe? 
*  Ed.  Encyc.,  Art.  Acoustic*. 


ACOUSTICS.  357 

not  form  an  element  of  the  algebraic  expression,  which  is  thus 
found  for  the  time  of  a  vibration,  it  follows  that  this  time  is 
independent  of  the  distance.  Hence,  as  in  the  pendulum, 

The  vibrations  of  a  string,  fixed  at  both  ends,  are  performed  in 
equal  times,  whether  the  length  of  the  vibrations  be  greater  or 
smaller. 

Upon  '  this  uniformity  in  the  times  of  vibration  depends  the 
uniformity  of  tone  ;  for  if  we  employ  a  string  of  unequal  thick- 
ness, and  consequently  one  whose  vibrations  are  performed  in 
different  times,  the  sound  is  confused  and  variable,  and  any  other 
mode  by  which  we  destroy  the  isochronism,  produces  a  similar 
effect.  The  same  law  has  been  found  to  extend  to  all  other  cases 
of  musical  sounds  ;  and,  therefore,  we  may  conclude,  that  isoch- 
ronism in  the  vibrations  of  sonorous  bodies,  is  essential  to  their 
producing  musical  sounds. 

The  number  of  vibrations  performed  by  a  string  in  a  second 
of  time,  being  inversely  as  the  time  of  one  vibration,  it  is  ex- 
pressed by  the  reciprocal  of  the  formula  denoting  the  time  ;  so 
that  if  N  represents  the  number  of  vibrations,  we  shall  have  the 


f  11  -AT 

following  expression  :  N  ~T   /in\' 

The  frequency  of  vibration  which  this  equation  gives,  is  found 
to  agree  very  exactly  with  the  result  of  experiments  performed 
with  strings,  whose  vibrations  are  so  slow  as  to  admit  of  being 
numbered. 

524.  The  relation  between  the  number  of  vibrations,  perform- 
ed by  different  strings,  may  be  expressed  by  a  more  simple  for- 
mula ;  for  g  and  the  two  numbers  being  constant  quantities,  they 
may,  in  this  case,  be  rejected,  and  we  get  the  following  expression  : 

N   oc-f  —  —  .     According,  then,  as  we  dimmish"  the  length  of  a 

LJ  v  U) 

string,  and  the  weight  of  an  inch  of  it,  or  increase  its  tension,  we 
increase  its  frequency  of  vibration  ;  but  equal  changes  in  these 
circumstances  do  not  produce  equal  effects.  Thus  if  in  different 
strings  their  tension  and  the  weight  of  an  inch  remain  the  same, 
their  frequency  of  vibration  will  be  inversely  as  their  lengths  ; 

for  then  N  oc  y-.  If  we  make  the  length  one  third,  we  triple  the 
number  of  vibrations,  and  so  for  any  other  proportion.  If  the 
length  and  tension  remain  the  same,  N  <x~7~t  or,  the  number  of 

vibrations  is  inversely  as  the  square  roots  of  the  weights  :  conse- 
quently a  string  four  times  as  heavy  as  another  will  vibrate  half 
as  fast.  The  bass  strings  in  most  instruments  have  fine  wire 
twisted  round  them  to  increase  their  quantity  of  matter,  other- 
wise greater  length  must  be  resorted  to  for  the  production  of  sim- 


358  NATURAL   PHILOSOPHY. 

ilar  tones.  If  the  length  and  the  weight  of  equal  portions  be  the 
same,  then  N  ocv/i,  or  the  frequency  of  vibration  is  as  the  square 
root  of  the  tension.  Therefore,  we  must  give  the  string  of  a  violin 
four  times  the  tension  in  order  to  make  it  vibrate  twice  as  fast. 

525.  WIND  INSTRUMENTS. — In  wind  instruments,  a  column  of 
confined  air  itself  is  the  vibrating  body ;  and  here  the  vibrations 
are  longitudinal  instead  of  lateral,  as  is  the  case  with  strings. 
That  it  is  really  the  air  which  is  the  sounding  body  in  a  flute,  or- 
gan pipe,  or  other  wind  instrument,  appears  from  the  fact,  that 
the  materials,  thickness,  or  peculiarities  of  the  pipe,  are  of  no 
consequence.     A  pipe  of  paper  and  one  of  lead,  glass,  or  wood, 
provided  the  dimensions  are  the  same,  produce,  under  similar 
circumstances,  exactly  the  same  tone  as  to  pitch.     If  the  qualities 
of  the  tones  produced  by  different  pipes  differ,  this  is  to  be  attribu- 
ted to  the  friction  of  the  air  within  them,  setting  in  feeble  vibra- 
tions their  own  proper  materials.*     The  class  of  bodies  vibrating 
longitudinally,  is  not  only  more  diversified  in  its  powers  than  the 
other  classes  of  sounding  bodies,  but  also  more  extensive  in  the 
range  of  substances  which  it  comprehends.     A  uniform  rod  under 
any  solid  substance,  or  a  column  of  air  contained  in  a  cylindrical 
tube,  whose  diameter  is  everywhere  equal,  may  have  its  vibra- 
tion limited  at  both  extremities  by  an  immovable  obstacle  ;  or 
both  extremities  may  be  at  liberty ;  or  one  extremity  may  be 
confined  and  the  other  disengaged. 

A  column  of  air,  or  a  rod  of  any  substance,  whether  confined 
or  free  at  both  extremities,  performs  a  double  vibration  in  the 
same  time  that  a  minute  impulse  would  occupy,  when  travelling 
in  a  medium  of  the  substance  through  twice  the  length  of  the 
sonorous  body ;  and  a  body  fixed  at  one  extremity  only,  will 
occupy  double  that  time.  Hence,  the  number  of  vibrations  per- 
formed in  a  secjond  of  time  by  a  given  body,  is  the  same,  wheth- 
er that  body  be  fixed  at  both  extremities,  or  free  at  both ;  and 
therefore  its  sound  in  these  two  cases  should  be  the  same.  But 
if  the  body  be  fixed  at  one  extremity  and  free  at  the  other,  its 
length  must  be  reduced  to  one  half,  to  make  it  give  the  same  tone 
as  in  the  two  former  cases.  Thus,  if  we  blow  into  a  tube  closed 
at  one  extremity,  it  will  give  the  same  tone  as  we  procure  by 
blowing  into  an  open  tube  of  double  the  length. 

526.  The  different  pitch  of  bodies  vibrating  longitudinally,  and 
free  at  both  extremities,  depends  on  four  circumstances,  viz.  their 
elasticity,  the  temporary  rate  at  which  their  elasticity  is  in- 
creased by  condensation,  their  length,  and  their  specific  gravity, 
the  tone  of  a  body  being  more  acute,  according  as  the  elasticity, 
and  the  rate  of  its  increase  by  condensation,  are  greater,  or  the 

*  Herschel. 


ACOUSTICS.  359 

length  and  specific  gravity  less.  The  length  of  the  sonorous 
body  is  almost  exclusively  the  only  one  of  these  circumstances 
which  we  have  completely  in  our  power ;  and  with  regard  to 
ordinary  wind  instruments,  and  all  musical  instruments  where 
common  air  is  the  vibrating  body,  the  length  is  the  circumstance 
of  most  importance,  since  the  elasticity,  rate  of  condensation,  and 
specific  gravity,  are  then  nearly  constant  quantities.  The  change 
of  specific  gravity,  however,  to  which  the  air  is  subject  in  conse- 
quence of  changes  of  temperature,  materially  affects  the  pitch 
of  wind  instruments.  The  frequency  of  vibration  of  a  column 
of  air  is  found  to  be  increased  about  3^,  by  an  elevation  of  30° 
Fahrenheit.  Thus,  the  tone  of  an  organ  has  been  found  to  be 
higher  in  summer  than  in  winter ;  and  flutes  and  other  wind  in- 
struments become  gradually  more  acute  as  the  included  air  is 
heated  by  the  breath.* 

527.  BELLS. — If  a  bell  be  struck  by  a  clapper  on  the  inside, 
the  bell  is  made  to  vibrate.     The  base  of  the  bell  is  a  circle  ;  but 
it  has  been  found  that,  by  striking  any  part  of  the  circle  on  the 
inside,  that  part  flies  out,  so  that  the  diameter  which  passes 
through  this  part  of  the  base,  will  be  longer  than  the  other  di- 
ameters.    The  base  is  changed  by  the  blow  into  the  figure  of 
an  ellipse,  whose  longer  axis  passes  through  the  part  against 
which  the  clapper  is  thrown.     The  elasticity  of  the  bell  restores 
the  figure  of  the  base,  and  again  elongates  the  bell  in  a  direction 
opposite  to  the  former  ;  and  the  two  elliptical  figures  thus  alter- 
nate with  each  other,  growing  smaller  and  smaller,  like  the  vi- 
brations of  a  pendulum  when  the  moving  force  is  withdrawn, 
until  the  sound  dies  away.     We  may  be  convinced  by  our  senses, 
that  the  parts  of  the  bell  are  in  a  vibratory  motion  while  it 
sounds.     If  we  lay  the  hand  gently  upon  it,  we  shall  feel  this 
tremulous  motion,  and  even  be  able  to  stop  it ;  or  if  small  pieces 
of  paper  be  put  upon  the  bell,  its  vibrations  will  set  them  in 
motion,  f 

We  may  conceive  the  bell  to  be  formed  of  an  infinitude  of 
rings,  placed  one  above  another,  from  the  base  to  the  highest 
point.  The  rings  situated  nearer  to  the  base,  having  a  greater 
circumference,  tend  to  perform  their  vibrations  more  slowly, 
while  the  rings  nearer  to  the  summit,  whose  circumferences  are 
smaller,  tend  to  produce  vibrations  oftener.  These  sounds  will 
so  coalesce  as  to  produce  a  mixed  sound,  intermediate  between 
those  of  the  higher  and  lower  rings.  J 

528.  THE  VOICE  AND  THE  EAR. — The  human  voice  depends  prin- 
cipally on  the  vibrations  of  the  membranes  of  the  glottis,  excited 
by  a  current  of  air  which  they  alternately  interrupt  and  suf- 

*  Ed.  Encyc.,  Art.  Acoustics.         t  Hatty's  Nat.  PhU.  I,  203.  \  Ib.  305. 


360  NATURAL   PHILOSOPHY. 

fer  to  pass  ;  the  sounds  being  also  modified  in  their  subsequent 
progress  through  the  mouth.* 

The  parts  of  the  ear,  and  the  progress  of  sound  to  the  sentient 
nerve,  may  be  simply  described  as  follows. 

(1.)  There  is  externally  a  wide-mouth-  Fig.  210. 

ed  tube  or  ear  trumpet,  a,  fo»*  catching 
and  concentrating  the  pulses  of  sound. 
In  many  animals  it  is  movable,  so  that 
they  can  direct  it  to  the  place  from  which 
the  sound  comes. 

(2.)  The  sound  concentrated  at  the  bot- 
tom of  the  ear  tube  falls  upon  a  mem- 
brane stretched  like  the  top  of  an  ordinary  drum,  over  the  tym 
panum  or  drum  of  the  ear,  b,  and  causes  it  to  vibrate.  That  its 
motion  may  be  free,  the  air  contained  within  the  drum  has  free 
communication  with  the  external  air  by  the  open  passage  f, 
called  the  Eustachian  tube,  leading  to  the  back  part  of  the  mouth. 
A  degree  of  deafness  ensues  when  this  tube  is  obstructed  by 
wax. 

(3.)  The  vibrations  of  the  tympanum  are  conveyed  further  in- 
wards by  a  chain  of  four  bones,  (not  here  represented  on  ac- 
count of  their  minuteness,)  reaching  from  the  center  of  the  tym- 
panum to  the  oval  door  or  window  of  the  labyrinth,  e. 

,(4.)  the  labyrinth,  or  complex  inner  compartment  of  the  ear, 
over  which  the  nerve  of  hearing  is  spread  as  a  lining,  is  full  of 
water  ;  and  therefore,  when  the  vibrations  of  the  tympanum 
acting  through  the  chain  of  bones  (3)  are  communicated  to  this 
fluid,  they  are  instantly  felt  over  the  whole  cavity.  (Art.  382.) 
The  labyrinth  consists  of  the  vestibule,  e,  the  three  semi-circular 
canals,  c,  imbedded  in  the  hard  bone,  and  of  a  winding  cavity,  d, 
called  the  cochlea,  like  that  of  a  snail  shell,  in  which  fibres, 
stretched  across  like  harp-strings,  constitute  the  lyra.  The  ex- 
act uses  of  these  various  parts  are  not  yet  perfectly  known. 
The  membrane  of  the  tympanum  may  be  pierced,  and  the  chain 
of  bones  may  be  broken,  without  loss  of  hearing,  f 


CHAPTER  II. 

OF  THE  PROPAGATION  OF  SOUND. 

529.  AIR  is,  in  general,  the  medium  of  sound.  A  bell  struck 
under  the  receiver  of  an  air-pump,  gives  a  feebler  and  feebler 
sound  as  the  exhaustion  proceeds,  until,  when  the  rarefaction  is 

»  Young's  Lectures,  I,  401.  t  Dr.  Arnott,  El.  Phys.,  Vol.  I,  p.  507. 


ACOUSTICS.  361 

carried  to  a  certain  extent,  it  emits  no  sound  at  all.*  Hence, 
on  the  summit  of  high  mountains,  where  the  air  is  naturally 
rare,  sound  ought  to  be  weaker  than  at  the  general  level  of  the 
earth  ;  and  such  is  found  to  be  the  fact.  Saussure  relates  that 
on  the  top  of  Mont  Blanc,  the  firing  of  a  pistol  made  a  report 
no  louder  than  that  of  a  child's  toy-gun.  A  fact  mentioned  by 
travellers  in  Alpine  countries,  is  explained  on  this  principle. 
They  see  distinctly  a  huntsman  on  a  neighboring  eminence,  and 
observe  the  flashes  of  his  gun,  but  can  scarcely  hear  the  report, 
even  when  comparatively  near  them.f 

Yet  meteoric  bodies  are  said  to  give  a  distinct  rumbling  sound 
in  passing  through  the  air  at  the  height  of  fifty  miles,  an  altitude 
at  which  the  air  is  rarefied  to  a  degree  exceeding  the  vacuum  of 
the  air-pump.  Dr.  Halley  mentions  an  instance  of  a  meteor, 
whose  elevation  was  at  least  sixty-nine  miles,  exploding  with  a 
sound  equal  to  "  the  report  of  a  very  great  cannon,  or  broadside." 
Probably,  however,  these  sounds  do  not  emanate  from  the  meteor 
itself,  but  from  fragments  projected  from  it,  which  fall  through 
the  air  to  the  ground.  If  the  "  rumbling  sound"  above  men- 
tioned, proceeds  from  the  body  of  the  meteor,  it  is  necessary  to 
suppose  that  the  air  is  condensed  before  it  to  a  great  extent.  On 
the  other  hand,  when  the  elasticity  of  the  air  is  augmented,  either 
by  condensation  or  by  heat,  the  force  of  sound  is  considerably 
increased.  This  effect  has  been  experienced  in  the  condensed 
air  of  diving  bells. 

530.  Air  receives  from  sounding  bodies  vibrations,  which  it  com- 
municates to  the  organs  of  hearing.  Let  us  take,  for  example,  a 
cord  of  a  stringed  instrument,  and  suppose  it  struck  when  played 
upon  ;  immediately  all  the  points  of  that  string  will  deviate 
more  or  less  from  the  position  which  they  occupied  when  the 
string  was  at  rest,  according  as  they  are  more  or  less  distant 
from  the  points  where  the  string  is  fixed  ;  and  the  string  will  go 
and  return  alternately  on  this  side  and  on  that  side  of  its  first 
situation,  by  a  vibratory  motion  occasioned  by  its  elasticity. 
The  particles  of  air  contiguous  to  the  different  points  of  the 
string,  assume  motions  similar  to  those  of  the  respective  points, 
that  is,  they  move  backward  and  forward  with  them.  Each 
particle  communicates  motion  to  that  which  is  next  to  it,  that  to 
a  third,  and  so  on,  until  the  particles  of  air  are  reached  which 
are  in  contact  with  the  tympanum,  or  drum  of  the  ear.  The 
air  then  acts  upon  that  membrane,  by  communicating  to  it  its 
own  vibrations,  which  the  drum  transmits  to  the  auditory  nerve ; 
and  thence  results  the  sensation  of  sound.J 

*  Herschel  in  Encyc.  Metrop.  II,  747.          t  Partington,  I.  263. 

t  Haiiy,  I,  203. — It  is  evident  from  the  mechanical  concussion  attending  loud 
fioises,  that  sound  consists  in  a  motion  of  the  air  itself,  communicated  along  it  by 
virtue  of  its  elasticity,  as  a  tremor  runs  along  a  stretched  rope. — Herschel  on  Sound. 
46 


362  NATURAL    PHILOSOPHY. 

531.  In  an  open  space,  and  in  a  serene  atmosphere,  sound  is 
propagated  from  the  sounding  body  in  all  directions.     Sounds, 
even  the  most  powerful,  when  thus  transmitted  freely  through 
the  air,  diminish  rapidly  in  force,  as  they  depart  from  their  sources, 
and  within  moderate  distances  wholly  die  away.     What  law  this 
diminution  follows,  is  not  yet  ascertained  ;  and  is  indeed,  in  the 
present  state  of  acoustics,  incapable  of  determination.     Some 
writers  have  supposed  that  sound  follows  the  common  law  of 
emanations  radiating  from  a  center,  (Art.  7,)  and  consequently, 
that  its  intensity  at  different  distances  from  its  source  varies  in- 
versely as  the  square  of  the  distance  ;*  but  we  can  estimate  the 
force  of  sounds  by  the  ear  alone  ;  an  instrument  of  comparison 
whose  decisions  on  this  point  vary  with  the  bodily  state  of  the 
observer,  and  whose  scale  expresses  no  definite  relation  but  that 
of  equality. 

Though  sound  has  in  general,  at  its  origin,  a  tendency  to  dif- 
fuse itself  in  all  directions,  it  is  sometimes  more  propagated  in 
one  direction  than  in  others.  A  cannon  seems  much  louder  to 
those  who  stand  immediately  before  it,  than  to  those  who  are 
placed  behind  it.  The  same  fact  is  illustrated  by  the  speaking 
trumpet ;  the  person  toward  whom  the  instrument  is  directed, 
hears  distinctly  the  words  spoken  through  it,  while  those  who 
are  situated  a  little  to  one  side,  hardly  perceive  any  sound. 

532.  Sound  is  in  a  great  measure  intercepted  by  the  interven- 
tion of  any  solid  obstacle  between  the  hearer  and  the  sonorous 
body.     Thus,  if  while  a  bell  is  sounding,  houses  intervene  be- 
tween us  and  the  bell,  we  hear  it  sound  but  faintly,  compared 
with  what  we  hear  after  we  have  turned  the  corner  of  the  build- 
ing.    From  this  fact  sound  would  seem  to  be  propagated  in 
straight  lines.     If,  however,  we  speak  through  a  tube,  the  voice 
will  be  wholly  confined  by  the  tube,  and  will  follow  its  windings 
however  tortuous  ;  hence  we  infer  that  sound  is  propagated  not 
in  right  lines  like  radiant  substances,  as  heat  and  light,  but  in 
undulations,  after  the  manner  of  waves,  such  as  follow  when  a 
stone  is  thrown  into  still  water. 

533.  Though  air  is  the  most  common  medium  of  sound,  yet 
it  is  not  the  only  medium.     Various  other  bodies,  both  solid  and 
fluid,  are  excellent  conductors  of  sound  ;  and  the  fainter  sound 
of  the  bell  when  buildings  intervene,  as  in  the  case  supposed, 
(Art.  532,)  arises  from  the  fact,  that  sound  passes  with  difficulty 
from  one  medium  into  another. 

If  a  log  of  wood  is  scratched  with  a  pin  at  one  extremity,  a 
person  who  applies  his  ear  to  the  othei;  extremity  will  hear  the 

*  Millington's  Nat.  Phil.  p.  125,  Epit.— Herschel  on  Sound,  Encyc.  MetropoL 
IIj  773 


ACOUSTICS.  063 

sound  distinctly ;  and  when  a  long  pole  of  wood  is  applied  at 
one  end  to  U»e  teeth,  the  ticking  of  a  watch  may  be  heard  at  the 
other  end,  at  a  much  greater  distance  than  when  there  is  no 
medium  of  communication  but  the  air.  The  motion  of  a  1  roop 
of  cavalry  is  heard  at  a  great  distance  by  applying  the  ear  close 
to  the  ground,  and  it  is  well  known  that  dogs,  by  this  method, 
first  discover  the  approach  of  a  stranger. 

534.  The  VELOCITY  of  sound  is  progressive.     Thus  when  a  gun 
is  fired  at  a  distance  from  us,  we  perceive  the  flash  some  time 
before  we  hear  the  report.     Thunder  follows  the  lightning  at  a 
perceptible  interval,  although  they  are  known  to  be  cotempora- 
neous  events.     If  a  gun  be  fired  at  a  certain  known  distance,  and 
we  observe  the  interval  between  the  flash  and  the  report,  we 
may  obtain  the  rate  at  which  sound  passes,  that  is,  the  velocity 
of  the  sound.     Many  years  since,  Dr.  Derham  made  a  number 
of  accurate  and  diversified  experiments  on  this  subject,  and  fixed 
the  velocity  of  sound  at  1142  feet  per  second.     The  mean  of  a 
great  number  of  experiments  gives  the  average  velocity  of  1130 
feet  per  second :    but  the  velocity  as  determined  by  Derham, 
namely,  1142  feet  per  second,  is  that  which  has  been  generally 
admitted  as  the  standard.     Since,  however,  the  transmission  of 
sound  depends  on  the  elasticity  of  the  medium,  (Art.  519,)  causes 
which  affect  the  elasticity,  likewise  affect  the  velocity  of  sound. 
Thus  the  velocity  is  a  little  greater  in  warm  than  in  cold  air, 
and    consequently  is    somewhat    influenced   by  climate.*      M. 
Goldingham,  by  a  series  of  experiments  made  at  Madras,  found 
that  the  velocity  of  sound  was  affected  even  by  the  seasons  of 
the  year,  increasing  regularly  from  the  coldest  to  the  hottest 
months,  and  afterward  regularly  decreasing.     Hence,  for  every 
degree  of  Fahrenheit's  thermometer  1.14  feet  is  allowed  for  the 
velocity  of  sound  per  second.     A  similar  gradation  in  the  ve- 
locity of  sound  at  different  seasons  of  the  year,  was  observed 
by  Capt.  Parry  in  his  experiments  on  sound  within  the  frigid 
zone.-f- 

All  sounds  travel  in  the  same  air  with  the  same  velocity,  other- 
wise there  could  be  no  such  thing  as  harmony,  an  essential  con- 
dition of  which  is  that  the  sounds  should  reach-  our  ears  in  a  pre- 
cise order,  and  at  exact  intervals.J 

535.  Sound  moves  with  a  uniform  velocity ;  that  is,  it  passes 
over  equal  spaces  in  equal  times.     This  important  fact  was  first 
ascertained  by  Derham,  who  found  that  it  held  good  whether  the 
sound  were  strong  or  feeble  ;  whether  it  proceeded  from  a  ham- 
mer or  a  cannon  ;  in  short,  that  neither  the  strength  nor  the  ori- 
gin of  the  sound  makes  any  difference.     M.  Biot  caused  several 

*  Partington,  I,  263.      t  Phil.  Trans.  1828,  p.  97.      t  Webster's  El.  Phys.  p.  194. 


364  NATURAL   PHILOSOPHY. 

airs  to  be  played  on  a  flute  at  the  end  of  an  iron  pipe  3120  feet 
long,  and  the  notes  were  distinctly  heard  by  him  at  the  other  end, 
without  the  slightest  derangement  in  the  order  or  quality  of  the 
sounds. 

The  velocity  of  seund,  however,  when  transmitted  through 
the  air,  is  slightly  influenced  by  the  strength  and  direction  of  the 
wind.  Dr.  Derham  found  that  when  the  wind  is  blowing  in  the 
direction  of  the  sound,  its  velocity  must  be  added  to  the  stand- 
ard velocity  of  sound,  and  must  be  subtracted  from  it  when  op- 
posed to  it.*  A  transverse  wind  does  not  affect  the  velocity  of 
sound  in  the  slightest  degree. 

536.  Several  distinguished  philosophers,  both  of  France  and 
Holland,  have  recently  made  experiments  on  the  velocity  of 
sound,  under  circumstances  the  most  favorable  to  the  attainment 
of  accurate  results.  A  difficulty  experienced  by  the  earlier  ex- 
perimenters, as  Derham,  arose  from  the  want  of  a  method  of  meas- 
uring a  small  fraction  of  a  second,  and  yet  this  was  necessary 
where  a  variation  of  one  hundredth  part  of  a  second  makes  a 
difference  of  more  than  eleven  feet  in  the  result.  The  Dutch 
experimenters!  employed  a  clock  with  a  conical  pendulum,  capa- 
ble of  determining  intervals  to  the  hundredth  of  a  second,  by 
suddenly  suspending  the  motion  of  the  index  without  stopping 
the  clock.  In  the  French  experiments  a  kind  of  watch  was  used, 
one  of  whose  hands  performed  a  revolution  in  a  second,  and  by 
the  sudden  pressure  of  a  small  lever  could  be  made  to  touch  with 
its  extremity  the  dial  plate,  al  any  instant,  and  leave  there  a  dot, 
without  interrupting  its  motion  or  rotation,  to  effect  which,  it 
carried  with  it  a  drop  of  printer's  ink,  in  a  peculiar  and  ingenious 
species  of  dotting  pen.  By  the  use  of  these  instruments,  it  was 
found  practicable  to  ascertain  the  interval  between  the  sight  of  a 
flash,  and  the  arrival  of  the  report  of  a  gun,  with  such  precision 
as  to  destroy  in  the  result  all  material  error  which  might  arise 
from  this  cause.  Accordingly,  their  results  afford  a  striking 
agreement, — the  experiments  of  the  French  gave  for  the  velocity 
of  sound,  per  second,  1086.1  feet:  those  of  the  Dutch,  1089.42, 
both  considering  the  air  at  the  temperature  of  freezing  water. 
But  it  is  found  that  the  velocity  is  increased  1.14  feet  for  every 
degree  of  Fahrenheit ;  consequently,  reducing  the  estimate  to  the 
temperature  of  62£°,  (which  is  the  standard  temperature  of  the 
British  metrical  system,)  the  velocity  becomes  1124.19,  as  deter- 

*  If  a  stone  be  thrown  into  a  small  lake,  the  waves  spread  with  equal  rapidity  in 
all  directions,  in  circles  whose  center  is  the  stone.  If  into  a  running  river,  still  they 
form  circles,  but  their  center  is  carried  down  the  stream ;  and,  in  point  of  fact,  the 
waves  arrive  opposite  to  the  point  of  the  bank  above  the  place  where  the  stone  fell, 
later  than  at  a  point  at  the  same  distance  below  it,  in  proportion  to  the  rapidity  of  the 
stream. — Herschel  on  Sound. 

t  Moll,  Vanbeck,  &c. 


X 

ACOUSTICS.  365 


mined  by  the  Dutch  experimenters,  which  is  deemed  the  most  ac 
curate  result  hitherto  obtained.  It  may,  therefore,  be  stated  in. 
round  numbers,  that  sound  passes  through  air  at  the  rate  of 
nine  thousand  feet  in  eight  seconds,  or  twelve  miles  and  three 
fourths  per  minute,  or  seven  hundred  and  sixty-five  miles  an  hour, 
which  is  about  three  fourths  of  the  diurnal  velocity  of  the  earth's 
equator.  Hence,  in  latitude  421°,  if  a  gun  were  fired  at  the  mo- 
ment a  star  passes  the  meridian  of  any  station,  the  sound  would 
reach  any  other  station  exactly  west  of  it  at  the  precise  instant  of 
the  same  star's  arriving  on  its  meridian  ;  that  is,  it  would  keep 
pace  with  the  velocity  of  the  earth  at  that  place,  as  it  turns  on 
its  axis,  in  the  diurnal  revolution.* 

From  a  knowledge  of  the  velocity  of  sound,  the  distance  of  a 
sounding  body  may  be  estimated.  Thus  if  the  interval  between 
seeing  a  flash  of  lightning,  and  hearing  the  thunder,  be  six  sec- 
onds, the  distance  of  the  cloud  is  6  x  1 130=6780  feet,  or  lr\  miles. 

537.  The  air  is  a  better  conductor  of  sound  when  humid  than 
when  dry.     Thus,  a  bell  is  heard  better  just  before  a  rain ;  and 
this  fact  lends  some  countenance  to  an  opinion  of  the  ancients, 
that  sound  is  heard  better  by  night  than  by  day.     Humboldt  was 
particularly  struck  with  this  fact,  when  he  heard  the  noise  of  the 
great  cataracts  of  Orinoco,  which  he  describes  as  three  times 
greater  in  the  night  than  in  the  day.f 

The  distance  to  which  sound  may  be  heard,  will  of  course 
vary  with  its  force  and  various  other  circumstances  which  are 
incapable  of  being  reduced  to  an  exact  law.  Volcanoes,  in  South 
America,  have  sometimes  been  heard  at  the  distance  of  three 
hundred  miles  ;  and  naval  engagements  have  been  heard  at  the 
distance  of  two  hundred  miles.  The  unassisted  human  voice 
has  been  heard  from  Old  to  New  Gibraltar,  a  distance  of  ten  or 
twelve  miles,  the  watch-word,  All's  Well,  given  at  the  former 
place  being  heard  at  the  latter.  Sounds  are  heard  to  a  much 
greater  distance  over  water  than  over  land,  and  further  on  smooth 
than  on  rough  surfaces. 

538.  Liquids  are  good  conductors  of  sound.     Indeed,  sound  is 
conveyed  with  far  greater  velocity  in  water  than  in  air.     Dr. 
Franklin,  having  plunged  his  head  below  water,  caused  a  person 
to  strike  two  stones  together  beneath  the  surface,  and  heard  the 
sound  distinctly  at  the  distance  of  more  than  half  a  mile.     By 
similar  experiments,  it  has  been  ascertained,  that,  though  water 
is  a  much  better  conductor  of  sound  than  air,  yet  the  sound  is 

*  Herschel  on  Sound,  in  Encyc.  Metrop.  II,  p.  751. 

t  Humboldt,  however,  accounts  for  the  greater  audibility  of  sounds  by  night  than 
by  day,  from  the  absence  of  those  ascending  and  descending  currents  of  air  which, 
while  the  sun  is  shining,  impair  the  uniformity  of  the  medium,  and  thus  diminish  its 
conducting  powers. 


366  NATURAL   PHILOSOPHY. 

greatly  enfeebled  by  passing  out  of  one  medium  into  the  other 
The  most  accurate  experiments  on  this  subject,  are  those  made 
in  the  year  1826  by  M.  Colladon,  in  the  Lake  of  Geneva.  He 
caused  a  bell  to  be  rung  under  water,  and  found  that  although 
the  sound  of  the  blow  was  well  heard  in  the  air  directly  above 
the  bell,  yet  the  intensity  of  the  sound  diminished  very  rapidly 
as  the  observer  removed  from  its  immediate  neighborhood,  and  at 
the  distance  of  two  or  three  hundred  yards,  it  could  no  longer  be 
heard  at  all.*  To  conduct  the  sound  from  the  water  to  the  ear, 
a  tin  pipe  was  employed,  which  was  plunged  into  the  water,  and 
the  ear  brought  close  to  the  upper  end.  By  this  contrivance  he 
was  enabled  to  hear  the  strokes  of  the  bell  in  water,  at  the  dis- 
tance of  about  nine  miles.  The  velocity  of  sound  under  water, 
M.  Colladon  found  to  be  four  thousand  seven  hundred  and  eight 
feet,  or  nearly  a  mile,  per  second.f 

539.  Solid  substances  convey  sound  with  various  degrees  of 
facility,  but  in  general  much  better  than  air,  and  as  well  or  even 
better  than  fluids.  By  placing  the  ear  against  a  long  dry  brick 
wall,  and  causing  a  person  at  a  considerable  distance  to  strike  it 
once  with  a  hammer,  the  sound  will  be  heard  twice,  because  the 
wall  will  convey  it  with  greater  rapidity  than  the  air,  though 
each  will  bring  it  to  the  ear.J  .  The  rate  at  which  cast  iron  con- 
ducts sound,  was  ascertained  by  M.  Biot  in  the  following  manner. 
He  availed  himself  of  the  laying  of  a  series  of  iron  pipes  to  con- 
vey water  to  Paris.  The  pipes  were  about  eight  feet  in  length, 
and  were  connected  together  with  small  leaden  rings.  A  bell 
being  suspended  within  the  cavity,  at  one  end  of  the  train  of 
pipes,  on  striking  the  clapper  at  the  same  instant  against  the 
side  of  the  bell,  and  against  the  inside  of  the  pipe,  two  dis- 
tinct sounds  were  successively  heard  by  an  observer  stationed 
at  the  other  extremity.  With  a  train  of  iron  pipes  two  thou- 
sand five  hundred  and  fifty  feet,  or  nearly  half  a  mile  in  length, 
the  interval  between  the  two  sounds  was  found  from  a  mean 
of  two  hundred  trials,  to  be  1.79  seconds.  But  the  transmis- 
sion of  sound  through  the  internal  column  6f  air,  would  have 
taken  2.2  seconds ;  which  shows  that  the  sound  occupied  only 
.41  of  a  second  in  passing  through  the  metal.  From  more  di- 
rect trials,  it  was  concluded  that  the  exact  interval  of  time,  du- 
ring which  the  sound  performed  its  passage  through  the  sub- 
stance of  the  train  of  pipes,  amounted  to  only  the  .26  of  a  sec- 


*  It  is  inferred  from  this  experiment,  that  sound  is  reflected  by  the  same  laws  as 
light ;  when  the  direction  is  perpendicular  to  the  reflecting  surface,  (in  this  case,  the 
surface  of  the  water,)  it  passes  without  reflexion ;  but  the  quantity  reflected  increases, 
as  the  angle  of  reflexion  is  more  oblique. 

t  Herschel. 

t  Millington,  p.  125. 


ACOUSTICS.  367 

end,  showing  that  iron  conducts  sound  about  ten  times  as  rapidly 
as  air  does.* 

If  a  string  be  tied  to  a  common  fire-shovel,  and  the  two  ends 
of  the  string  be  wound  around  the  fore  fingers  of  each  hand,  and 
the  fingers  be  placed  in  the  ears,  on  striking  the  bottom  of  the 
shovel  against  an  andiron  or  other  solid  body,  very  deep  and 
heavy  tones  will  be  heard,  and  the  vibrations  of  the  metal  will 
be  clearly  perceived. 

540.  Solids,  as  well  as  other  bodies,  owe  their  power  of  con- 
ducting sound  to  their  elasticity.  By  elasticity  in  a  solid,  how- 
ever, is  not  meant  a  power  of  undergoing  great  extensions  and 
compressions,  after  the  manner  of  air,  or  India-rubber,  and  return- 
ing readily  to  its  former  dimensions ;  but  rather  what  is  common- 
ly called  hardness,  in  contradistinction  to  toughness,  a  violent 
resistance  to  the  displacement  of  its  molecules  inter  se  in  all 
directions.  Thus  the  hardest  solids  are,  in  general,  the  most 
elastic,  as  glass,  steel,  and  the  hard  brittle  alloys  of  copper  and 
tin,  and  in  proportion  as  they  are  elastic,  they  are  adapted  to  the 
free  propagation  of  sound  through  their  substance.  But  an  im- 
portant condition  in  their  constitution  is,  that  their  substance  be 
homogeneous,  and  their  structure  uniform.  By  the  want  of  ho- 
mogeneity and  uniformity  in  the  conducting  medium,  the  sono- 
rous pulses  are  every  instant  changing  their  medium,  and  the 
general  wave  is  broken  up  into  a  multitude  of  non-coincident 
waves,  emanating  from  different  origins,  and  crossing  and  inter- 
fering with  each  other  in  all  directions.  Thus,  a  glass  vessel 
containing  an  effervescing  liquor,  cannot  be  made  to  ring,  but 
gives  a  dead  sound ;  but  as  the  effervescence  subsides,  the  tone 
becomes  clearer,  and  when  the  liquid  is  perfectly  tranquil,  the 
glass  rings  as  us\ial.f 

The  great  power  of  solid  bodies  to  conduct  sound  is  exempli- 
fied in  earthquakes,  which  are  heard  almost  simultaneously  in 
very  distant  parts  of  the  earth.  Musical  boxes  sound  much 
louder  when  placed  on  a  table  or  some  solid  support,  than  when 
the  air  affords  the  only  conducting  medium.  It  is  easy  to  ascer- 
tain whether  a  kettle  boils,  by  putting  one  end  of  a  stick  or  poker 
on  the  lid,  and  the  other  end  to  the  ear :  the  bubbling  of  the 

*  Herschel  observes,  that  from  this  determination  we  may  estimate  the  time  it 
requires  to  transmit  force,  (whether  by  pulling,  pushing,  or  by  a  blow,)  to  any  dis- 
tance, by  means  of  iron  bars  or  chains.  For  every  eleven  thousand  and  ninety  feet 
of  distance,  (=the  velocity  of  sound  per  second  in  iron,)  the  pull,  push,  or  blow  will 
reach  its  point  of  action  one  second  after  the  moment  of  its  first  emanation  from  the 
first  mover.  In  all  moderate  distances,  then,  the  interval  is  utterly  insensible.  But, 
were  the  sun  and  the  earth  connected  by  an  iron  bar,  no  less  than  one  thousand  and 
seventy-four  days,  or  nearly  three  years,  must  elapse  before  a  force  applied  at  the  sun 
could  reach  the  earth.  The  force  actually  exerted  by  their  mutual  gravity  may  be 
proved  to  require  no  appreciable  time  for  its  transmission.  H  jw  wonderful  is  this  con. 
nection  ! — Herschel  on  Sound,  Encyc.  Metrop.  II,  773. 

t  Herschel. 


368  NATURAL   PHILOSOPHY. 

water,  when  it  boils,  appears  louder  than  the  rattling  of  a  car- 
riage in  the  streets.  A  slight  blow  given  to  the  poker,  of  which 
the  end  is  held  to  the  ear,  produces  a  sound  which  is  even  pain- 
fully loud.* 

A  physician  of  Paris  introduced  into  medical  practice  an  in- 
strument, depending  on  the  power  of  solid  bodies  to  conduct 
sound,  called  the  Stethoscope,^  the  object  of  which  is  to  render 
audible  the  action  of  the  heart  and  the  neighboring  organs.  It 
consists  of  a  wooden  cylinder,  one  end  of  which  is  applied  firmly 
to  the  breast  of  the  patient,  while  the  other  end  is  brought  to 
the  ear.  By  this  means,  the  processes  that  are  going  on  in  the 
organs  of  respiration,  and  in  the  large  blood-vessels  about  the 
heart,  may  be  distinctly  heard  ;  and  it  is  said  that  the  stethoscope, 
when  skilfully  used,  "  becomes  the  means  of  ascertaining  some 
diseases  in  the  chest,  almost  as  effectually  as  if  there  were  con- 
venient windows  for  visual  inspection."! 


CHAPTER  III. 

OF  THE  REFLEXION  OF  SOUND. 

541.  SOUND  is  reflected  by  hard  bodies,  producing  the  well  known 
phenomenon  called  an  ECHO.     If  a  straight  line  be  drawn  from 
the  sounding  body  to  the  reflecting  surface,  representing  the 
course  of  the  sound  before  reflexion,  and  another  straight  line  be 
drawn  from  the  reflecting  surface,  in  the  direction  of  the  sound 
after  reflexion,  these  two  lines  will  make  equal  angles  with  that 
surface ;  that  is,  when  sound  is  reflected,  the  angle  of  reflexion 
is  equal  to  the  angle  of  incidence. 

The  surfaces  of  various  bodies,  solids  as  well  as  fluids,  have 
been  found  capable  of  reflecting  sounds,  viz.  the  sides  of  hills, 
houses,  rocks,  banks  of  earth,  the  large  trunks  of  trees,  the  surface 
of  water,  especially  at  the  bottom  of  a  well,  and  sometimes  even 
the  clouds.  §  It  is  therefore  evident  that  in  an  extensive  plain, 
or  at  sea,  where  there  is  no  elevated  body  capable  of  reflecting 
sounds,  no  echo  can  be  heard.  It  is  hence  easy  to  see  why  the 
poets,  who  convert  echo  into  an  animated  being,  place  her  habi- 
tation near  mountains,  rocks,  and  woods.|| 

542.  An  echo  is  heard  when  a  person  stands  in  a  position  to 
hear  both  the  original  and  the  reflected  sound ;  and  the  interval 


Arnott's  El.  Phys.  I,  497.  t  orHOos,  the  chest ;  axoviu,  tc  examine. 

Dr.  Arnot.t  §  Cavallo,  II,  345.  U  Haxiy. 


ACOUSTICS.  369 

will  be  greater  or  less  according  to  the  distance  of  the  reflecting 
surface  from  the  sounding  body  and  from  the  hearer,  and  hence 
the  interval  may  be  made  a  measure  of  the  distance.  If  the 
sound  of  the  voice  returns  to  the  speaker  in  two  seconds,  the  dis- 
tance of  the  reflecting  surface  is  one  thousand  one  hundred  and 
thirty  feet,  and  in  that  proportion  for  other  intervals.  Thus, 
the  breadth  of  a  river  may  be  ascertained  when  there  is  an  echo- 
ing rock  on  the  further  shore.  A  perpendicular  mountain's  side, 
or  lofty  cliffs,  such  as  frequently  skirt  the  sea  coast,  sometimes 
return  an  echo  of  the  discharge  of  artillery,  or  of  a  clap  of  thun- 
der, to  the  distance  of  many  miles.*1  The  number  of  syllables 
that  can  be  pronounced  in  half  the  interval,  will  be  repeated 
distinctly ;  but  a  greater  number  would  be  blended  with  the 
commencement  of  the  echo.f 

When  a  single  obstacle  reflects  the  sound,  the  echo  is  simple  ; 
when  there  are  several  obstacles  disposed  at  suitable  distances, 
the  echo  is  complex.  Echoes  of  the  latter  kind  have  been  ob- 
served which  repeated  the  original  sound  forty  times.J  Two 
parallel  walls  which  mutually  reverberate  the  sound,  may 
produce  a  double  or  complex  echo,  with  regard  to  an  auditor 
placed  in  the  intermediate  space.  The  sound  of  artillery  and  of 
thunder,  is  frequently  prolonged  by  reverberations  in  an  uneven 
country. 

543.  The  furniture  of  a  room,  especially  the  softer  kind,  such 
as  curtains  or  carpets,  impair  the  qualities  of  sound  by  presenting 
surfaces  unfavorable  to  vibrations.  A  crowded  audience  has  a 
similar  effect  and  increases  the  difficulty  of  speaking.  Halls  for 
music  or  declamation,  should  be  constructed  writh  plain  bare 
walls.  Alcoves,  recesses,  and  vaulted  ceilings,  produce  reverbe- 
rations which  often  greatly  impair  the  distinctness  of  elocution. 
Indeed,  the  qualities  of  a  room,  in  regard  to  sound,  are  modified 
by  so  many  circumstances,§  that  the  science  of  acoustics  is 
worthy  of  more  attention  from  the  architect  than  it  has  generally 
received.  Plane  and  smooth  surfaces  reflect  sound  without 
dispersing  it,  convex  surfaces  disperse  it,  and  concave  surfaces 
collect  it.  The  concentration  of  sound  by  concave  surfaces, 
produces  many  curious  effects  both  in  nature  and  art.  There 
are  remarkable  situations  where  the  sound  from  a  cascade  is 
concentrated  by  the  surface  of  a  neighboring  cave,  so  completely, 

*  Arnott.  t  Cavallo,  II,  347.  \  Haiiy. 

§  The  famous  Dr.  Sanderson,  formerly  Professor  of  Mathematics  in  the  University 
of  Cambridge,  who  had  been  blind  from  the  time  he  was  a  year  old,  possessed  such 
acuteness  of  hearing,  that  he  not  only  distinguished  persons  with  whom  he  had  ever 
once  conversed,  so  long  as  to  fix  in  his  memory  the  sound  of  their  voice,  but  he  could 
also  recognise  places  by  observing  the  manner  in  which  they  modified  sound.  He 
could  judge  accurately  of  the  size  of  a  room,  and  of  his  distance  fi^m  the  wall ;  and 
if  ever  he  had  walked  over  a  pavement  in  courts,  or  piazzas,  and  was  conducted 
thither  agjiin,  he  could  tell  his  exact  situation,  by  the  note  which  the  place  sounded. 

47 


370  NATURAL    PHILOSOPHY. 

that  a  person  accidentally  bringing  his  ear  into  the  focus,  is 
astounded  by  a  deafening  noise.  Sound  issuing  from  the  center 
of  a  circle  is,  by  reflexion,  returned  to  the  center  again,  pro- 
ducing a  very  powerful  echo.*  Such  effects  are  observed  in 
the  central  parts  of  a  circular  hall.  An  elliptical  apartment 
conveys  sound  very  perfectly  from  one  focus  to  the  other.  A 
whisper  uttered  by  a  person  in  one  focus  of  such  a  chamber 
will  be  audible  to  a  person  in  the  other  focus,  though  not  heard 
by  persons  between. 

544.  Whispering  Galleries^  are  constructed  on  this  principle. 
Domes,  as  that  of  St.  Paul's  Cathedral,  in  London,  sometimes 
exhibit  the  same  curious  property. J     Concave  surfaces  facing 
each  other,  as  two  alcoves  in  a  garden,  or  covered  recesses  on 
opposite  sides  of  a  street,  or  bridge,  will  enable  persons  seated 
in  their  foci  to  converse  by  whispers,  notwithstanding  louder 
noises  in  the  space  between,  and  without  themselves  being  over- 
heard in  that  space.§     A  notorious  instance  of  a  sound-collecting 
surface,  was  the  ear  of  Dionysius,  in  the  dungeons  of  Syracuse. 
The  roof  of  the  prison  was  so  formed  as  to  collect  the  words, 
and  even  whispers  of  the  unhappy  prisoners,  and  to  direct  them 
along  a  hidden  conduit  to  the  place  where  the  tyrant  sat  listen- 
ing.    The  wide-spread  sail  of  a  ship,  rendered  concave  by  a  gen- 
tle breeze,  is  also  a  good  collector  of  sound.     Dr.  Arnott  relates 
an  instance  where  the  ringing  of  the  bells  at  St.  Salvador  on  the 
coast  of  Brazil,  was  heard  on  board  a  ship  at  the  distance  of  one 
hundred  miles  from  land.|| 

545.  The  most  frequent  instances  of  the  reflexion  of  sound, 
are  from  surfaces  which  may  be  considered  as  plane.     In  these, 
tjie  sound  issuing  from  any  point  seems,  after  reflexion,  to  proceed 
from  a  point  equally  distant,  and  similarly  situated,  on  the  other 
side  of  the  reflecting  surface  ;  the  phenomena  differing  a  little 
according  to  the  position  of  the  speaker,  with  respect  to  the  body 
which  occasions  the  reflexion.     If  a  person's  voice  strike  any 
surface  perpendicularly,  it  will  be  reflected  back  in  the  same 
line  ;  "and  the  time  occupied  between  the  utterance  of  the  sound, 
and  its  arrival  again  at  the  speaker,  will  be  equal  to  the  time  in 
which  the  sound  travels  through  twice  the  distance  between  the 
speaker  and  the  reflecting  surface.     The  interval,  therefore,  be- 

*  If  a  spherical  room  ceiild  be  constructed  of  perfectly  solid  materials,  perfectly 
polished,  aud  a  sound  were  to  issue  from  the  voice  of  a  person  in  the  center,  there 
would  be  an  accumulation  of  echo  at  the  center,  which  would  probably  be  destructive 
of  the  organs  of  hearing. — Latrobe  in  Ed.  Encyc. 

t  The  Hall  of  Secrets,  as  it  is  called,  in  the  Observatory  at  Paris,  is  a  whispering 
gallery.  This  hall  is  of  an  octagonal  form,  with  cloister  arches,  or  arched  by  portions 
of  a  cylinder,  which  meet  at  angles,  corresponding  to  those  formed  by  the  sides  of  the 
building.  The  speaker  applies  his  mouth  very  near  to  the  wall  to  one  of  the  angles, 
and  the  person  situated  at  the  opposite  angle  hears  his  voice  distinctly. 

1  Cavallo.  §  Arnott.  ||  El.  Phys.  I,  505. 


ACOUSTICS.  371 

tween  setting  out  and  returning,  will  be  found  by  the  following 
rule  :  Let  x  =  the  interval  in  seconds,  and  d  —  twice  the  dis- 
tance from  the  sounding  body  to  the  reflecting  surface  ;  then 

I  :  1130  ::  x  :  d  .'.  x  —  TTT     I£  therefore,  the  distance  is  less 


than  forty-eight  feet,  the  interval  of  time  between  the  speaker's 
hearing  the  direct  and  the  reflected  sounds,  will  be  less  than  TV 
of  a  second,  and  the  two  sounds  will  seem  to  coalesce  and  form 
but  one  sound  ;  but  if  the  distance  exceeds  forty-eight  feet,  then 
the  interval  will  be  greater  than  TV  of  a  second,  and  as  this  in- 
terval can  be  discerned  by  the  ear,  the  two  sounds  will  be  sepa- 
rate, and  will  form  an  echo.* 

546.  The  rolling  of  thunder  has   been  attributed  to  echoes 
among  the  clouds  ;  and  that  such  is  the  case  has  been  ascer- 
tained, by  direct  observation  on  the  sound  of  a  cannon.     Under 
a  perfectly  clear  sky,  the  explosion  of  guns  is  heard  single  and 
sharp  ;  while  when  the  sky  is  overcast,  or  when  a  large  cloud 
comes  overhead,  the  reports  are  accompanied  by  a  continued 
roll,  like  thunder,  and  occasionally  a  double  report  arises  from  a 
single  shot.f 

The  continued  sound  of  distant  thunder,  which  is  sometimes 
prolonged  for  many  seconds,  is  not  always  owing  to  reverbera- 
tion, but  frequently  arises  simply  from  the  different  distances  of 
the  same  flash.  Although  the  progress  of  a  flash  of  lightning 
through  the  air  were  absolutely  instantaneous,  still,  if  its  path 
were  in  a  line  that  would  carry  it  further  from  the  ear  in  one 
place  than  in  another,  there  would  be  a  corresponding  difference 
in  the  times  at  which  the  sound  generated  in  different  portions 
of  the  path  would  reach  the  ear.  Herschel  observes  that  if  (as 
is  almost  always  the  case)  the  flash  be  zigzag,  and  composed  of 
broken  rectilinear  and  curvilinear  portions,  some  concave,  some 
convex  to  the  ear,  —  and  especially  if  the  principal  trunk  sepa- 
rates into  many  branches,  each  breaking  its  own  way  through 
the  air,  and  each  becoming  a  separate  source  of  thunder,  —  all 
the  varieties  of  that  awful  sound  are  easily  accounted  for.J 

547.  The  Speaking  Trumpet  has  been  supposed  by  most  wri- 
ters on  sound,  to  owe  its  peculiar  properties  to  its  multiplying 
sound  by  numerous  reflexions.     Hence  is  suggested  the  form  of 
a  parabolic  conoid,  or  a  tube  the  section  of  which  is  a  parabola, 
the  place  of  the  mouth  being  at  the  focus  of  the  parabola.     The 
vibrations  emanating  from  the  mouth  would  then  be  reflected 
into  straight  lines  parallel  with  the  axis  of  the  trumpet,  and 

*  Edinburgh  Encyc.,  Art.  Acoustics.  t  Herschel. 

t  Herschel  on  Sound,  Encyc.  Metrop.,  II,  754. 


372  NATURAL   PHILOSOPHY. 

would  thus  go  forward  in  a  collected  body  to  a  distant  point.* 
And,  since  such  a  form  is  also  favorable  for  collecting  distinct 
sounds  into  one  point,  the  same  figure  is  proposed  as  the  most 
suitable  for  the  Ear  Trumpet.  But  the  sound  of  these  instru- 
ments may  be  regarded  as  merely  the  longitudinal  vibration 
(Art.  525)  of  a  body  of  air,  to  which  momentum  is  given  in  the 
direction  of  the  axis,  not  by  reflexion  from  the  sides,  but  by  the 
direct  impulse  of  the  mouth. f  The  ancients  were  acquainted 
with  the  speaking  trumpet.  Alexander  the  Great  is  said  to  have 
had  a  horn,  by  means  of  which  he  could  give  orders  to  his  whole 
army  at  once.J 

548.  Sound  may  be  conveyed  to  a  much  greater  distance  by 
being  confined,  during  its  whole  transmission,  within  a  pipe. 
Pipes  used  for  this  purpose  are  called  Acoustic  Tubes.     Such 
tubes  are  frequently  employed  in  public  houses  for  conveying 
orders  to  the  attendants.     Dr.  Herschel  employed  a  similar  tube, 
attached  to  his  forty  feet  telescope,  for  communicating  his  obser- 
vations to  an  assistant  who  sat  in  a  small  house  near  the  instru- 
ment, and  thus,  under  cover,  noted  them  down,  and  the  particu- 
lar time  in  which  they  were  made.     Acoustic  tubes  are  com- 
monly of  a  cylindrical    form,  and  have  at  each  extremity  a 
mouthpipe  like  that  of  a  speaking  trumpet,  to  which  either  the 
mouth  or  ear  is  applied,  according  as  the  person  is  speaking  or 
listening  to  another.     In  the  deception  called  the  Invisible  Girl, 
the  sound  of  the  voice  is  transmitted  and  returned  through 
acoustic  tubes. 

549.  Ventriloquism  does  not,  as  is  frequently  supposed,  depend 
on  the  reflexion  of  sound,  but  wholly  on  the  inaccuracy  with 
which  the  ear  judges  of  the  direction  from  which  sound  pro- 
ceeds,— enabling  the  performer,  by  a  variation  of  his  tone  of 
voice,  and  by  seeming  not  to  move  his  lips,  to  persuade  the  spec- 
tators that  the  sound  proceeds  from  some  object  to  which  he  has 
directed  their  attention.     The  imitations  of  different  sounds  by 
which  the  ventriloquist  is  able  to  personate  a  variety  of  charac- 
ters, and  to  represent  them  as  engaged  in  an  animated  dialogue 
with  each  other,  are  usually  limited  to  a  comparatively  small 
number,  which  have  been  acquired  and  rendered  very  familiar 
by  long  practice.     Hence,  like  the  performer  on  a  musical  in- 
strument, he  makes  his  transitions  from  one  sound  to  another 
with  a  facility  which  can  be  acquired  only  by  force  of  habit. 

550.  Sounding  Boards  were  formerly  constructed  over   the 
desks  of  public  speakers,  particularly  in  churches,  with  the  view 
of  aiding  the  powers  of  the  voice.     Their  efficacy  depended  on 

*  Dr.  Young,  Nat.  Phil.,  I,  375.  t  Ed.  Encyc.  II,  118. 

t  Enfield's  Sclent.  Rec.,  p.  157 


ACOUSTICS.  373 

the  reflexion  of  the  sound  ;  for  being  near  the  speaker,  the  echo 
or  reflected  sound,  uniting  itself  with  the  direct  sound,  would 
augment  its  force  or  loudness.  In  stringed  instruments,  however, 
as  the  violin,  the  sounding  board  acts  by  receiving  vibrations 
from  the  string.  Thus  by  impelling  the  air  with  a  greater  sur- 
face, it  produces  a  more  powerful  sound  than  the  string  alone. 
Hence,  if  some  weight,  (called  a  mute,}  as  a  penknife  partly  open, 
is  attached  to  the  bridge  of  a  violin,  the  sound  is  greatly  dead- 
ened, the  vibrations  of  the  string  being  thus  prevented  from  ex- 
tending to  the  sounding  board.* 

The  concave,  undulating,  and  perfectly  polished  surface  of 
many  sea  shells,  fits  them  to  catch,  to  concentrate,  and  to  return 
the  pulses  of  all  sounds  that  happen  to  be  trembling  about  them, 
so  as  to  produce  that  curious  resonance  from  within,  which  re- 
sembles the  distant  murmur  of  the  ocean,  f  The  organs  of 
speech  and  of  hearing  have  a  mechanical  structure  most  skil- 
fully adapted  to  the  peculiar  nature  of  sound. 

^_ 
CHAPTER  IV. 

OF  THE  PHILOSOPHICAL  PRINCIPLES  OF  MUSIC. 

551.  ON  this  subject,  we  have  room  for  only  a  few  leading 
principles. 

When  separate  sounds  are  repeated  with  a  certain  degree  of 
frequency,  the  ear  loses  the  power  of  distinguishing  the  intervals, 
and  they  appear  united  in  one  continued  sound.  By  this  means 
also,  sounds  harsh  and  dissonant  in  themselves,  form  a  soft  and 
agreeable  tone.  Any  sound  whatever,  repeated  not  less  than 
thirty  or  forty  times  in  a  second,  excites  in  the  hearer  the  sensa- 
tion of  a  musical  note.  Nothing  is  more  unlike  a  musical  sound 
than  that  of  a  quill  drawn  slowly  across  the  teeth  of  a  coarse 
comb ;  but  when  the  quill  is  applied  to  the  teeth  of  a  wheel 
whirling  at  such  a  r#te  that  720  teeth  pass  under  the  quill  in  a 
second,  a  very  soft,  clear  note  is  heard.J  In  like  manner  the 
vibrations  of  a  long  harp-string,  while  it  is  very  slack,  are  sepa- 
rately visible,  and  the  pulses  produced  by  it  in  the  air  are  sepa- 
rately audible  ;  but  as  it  is  gradually  tightened,  its  vibrations 
quicken,  and  the  eye  soon  sees,  when  it  is  moving,  only  a  broad 
shadowy  plain ;  the  distinct  sounds  which  the  ear  lately  perceived, 
run  together,  owing  to  the  shortness  of  the  intervals,  and  are 
heard  as  one  uniform  continued  tone,  which  constitutes  the  note 
or  sound  proper  to  the  string.^ 

Nature  presents  us  with  numerous  examples  of  a  musical 

*  Ed.  Encyc.,  II,  119.  t  Arnott. 

\  Robison's  Mech.  Phil.  IV,  404.  §  Arnott 


374  NATURAL   PHILOSOPHY. 

sound  produced  by  the  rapid  succession  of  an  individual  sound, 
not  at  all  musical  in  itself.  The  hum  of  winged  insects,  pro- 
duced by  the  frequent  motion  of  their  wings — the  murmur  of  a 
forest,  occasioned  by  the  agitation  of  the  leaves  and  boughs — 
and  the  sublime  roar  of  the  ocean,  constituted  of  the  separate 
sounds  produced  by  innumerable  waves,  are  familiar  examples 
of  the  operations  of  this  principle. 

552.  Musical  intervals,  or  sounds  differing  from  each  other  in 
pitch  by  a  certain  interval,  are  found  by  experience  to  be  pecu- 
liarly agreeable  to  the  human  ear  ;  a  fact  for  which  we  can  as- 
sign no  reason,  except  that  such  is  the  constitution  of  the  mind.* 

Musical  sounds  have  certain  ratios  to  one  another,  and  are 
thus  brought  within  the  province  of  Mathematics,  because  the 
number  of  vibrations  which  produce  one  musical  note,  has  a 
constant  ratio  to  the  number  which  produces  another  musical 
note.  Thus,  if  we  diminish  the  length  of  a  musical  string  one 
half,  we  double  the  number  of  its  vibrations  in  a  given  time, 
(Art.  524,)  and  it  gives  a  sound  eight  notes  higher  in  the  scale 
than  that  given  by  the  whole  string.  Therefore,  these  sounds 
are  represented  by  the  numbers  2  and  1,  and  are  said  to  be  in 
the  ratio  of  2  to  1.  The  upper  note  is  said  to  be  the  octave  of 
the  lower  ;  and  from  its  great  resemblance  to  the  fundamental 
note,  or  that  afforded  by  the  whole  string,  it  is  considered  as  the 
commencement  of  a  repetition  of  the  same  series ;  so  that  all 
audible  sounds  are  considered  as  repetitions  of  a  series  contained 
within  the  interval  of  an  octave.f 

553.  The  length  of  the  entire  string  being  called  1,  the  re- 
spective lengths  of  the  strings  which  sound  the  eight  notes,  are 
I » l>  4'  f >  f  >  T8s>  £•     The  sound  given  by  the  whole  string,  which 
is  denoted  by  1,  is  called  the  key  note,  and  the  other  notes  are 
called,  respectively,  the  second,  third,  fourth,  fifth,  sixth,  seventh, 
and  eighth,  and  the  fractions  denote  the  relation  of  each  note  in 
the  scale  to  the  key  note.     Since  the  number  of  vibrations  is  in- 
versely as  the  length  of  the  string,  (Art.  524,)  these  fractions  in- 
verted will  express  the  number  of  vibrations  which  produce  the 
several  notes  of  the  scale  respectively.     Thus,  f  denotes  that  the 
string  which  sounds  the  next  note  above  the  key  note  vibrates 
9  times,  while  the  whole  string  vibrates  8  times.     Hence,  the 
series  expressing  the  number  of  vibrations  which  produce  the 
notes  of  the  scale,  are  1,  f,  f,  f,  f,  f,  V,  2. 

But,  on  reducing  these  numbers  to  a  common  denominator, 
and  taking  their  numerators,  (which  express  the  ratios  of  the 
fractions,):];  we  have  the  following  series,  24,  27,  30,  32,  36,  40, 
45,  48. 

*  Playfair's  Outlines,  I,  274.  t  Young,  II,  393.  t  Day's  Algebra,  Art.  360, 
Cor.  1. 


ACOUSTICS.  375 

Hence  we  have  the  following  proposition  : 

If  a  string  be  divided,  so  that  the  number  of  vibrations  performed 
by  each  part  in  a  given  time,  shall  be  in  the  ratio  respectively  of  the 
numbers  24,  27,  30,  32,  36,  40,  45,  48,  the  sounds  of  the  first  seven 
will  be  perceived  as  increasing  in  acuteness  one  above  another,  from 
the  first  to  the  last,  and  will  yield  the  notes  from  the  combinations  of 
which  all  musical  effects  are  produced* 

554.  By  inspecting  the  last  series  of  numbers,  namely,  that 
which  expresses  the  relation  between  the  successive  notes  of  the 
diatonic  scale,  we  shall  perceive  that  the  ratios  between  two 
successive  numbers,  and  of  course  the  intervals  between  the 
several  notes  of  the  scale,  are  not  all  equal  to  each  other. 

1.  The  ratio  of  27  to  24  is  that  of  9:8 

2.  "  30  to  27  "  10:9 

3.  "  32  to  30  "  16 :  15 

4.  "  36  to  32  "  9:8 

5.  «     "  40  to  36  "  10 :     9 

6.  "  45  to  40  "  9:8 

7.  "  48  to  45  "  16:  15 
Hence  it  appears  that  there  are  in  the  musical  scale  three  sorts 

of  intervals,  of  which  three  bear  to  the  fundamental  or  key  note 
the  ratio  of  9  to  8,  two  that  of  10  to  9,  and  two  more  that  of  16 
to  15.  The  first  of  these  intervals  being  the  largest,  is  denom- 
inated the  major  tone,  the  second  the  minor  tone,  and  the  third 
the  semitone.  The  scale  therefore  is  made  up  of  three  major, 
two  minor,  and  two  semitones,  as  represented  in  the  table. 

After  ascending  through  the  first  seven  notes  of  the  scale,  we 
arrive,  as  has  been  already  intimated,  (Art.  552,)  at  a  note  which 
seems  to  be  only  a  repetition  of  the  first ;  hence  it  commences  a 
new  series  of  seven  notes  analogous  to  the  former  series,  each 
note  being  an  octave  above  the  corresponding  note  in  that  series, 
and  therefore  implying  vibrations  twice  as  rapid.  A  third  series 
is  constituted  in  the  same  manner,  called  the  double  octave,  in 
which  the  lengths  of  the  string  are  £  of  those  in  the  first  part  of 
the  scale. 

All  musical  sounds  are  computed  to  be  contained  between  ten 
octaves ;  so  that  the  number  of  vibrations  in  a  given  time  that 
yields  the  gravest  note,  is  to  that  which  yields  the  most  acute,  as 
1  to  210,  that  is,  as  1  :  1024.f 

555.  When  the  vibrations  are  less  numerous  than  about  16 
per  second,  the  ear  loses  the  impression  of  a  continued  sound, 
and  perceives,  first,  a  fluttering  noise,  then  a  quick  rattle,  then  a 
succession  of  distinct  sounds  capable  of  being  counted.     On  the 
other  hand,  when  the  frequency  of  the  vibrations  exceeds  a  cer- 

*  Playfair'a  Outlines,  I,  273.  t  Ib,  274. 


876  NATURAL    PHILOSOPHY. 

tain  limit,  all  sense  of  pitch  is  lost ;  a  shrill  squeak,  or  chirp,  only 
is  heard ;  and  what  is  very  remarkable,  many  individuals,  no 
way  inclined  to  deafness,  are  altogether  insensible  to  very  acute 
sounds,  even  such  as  painfully  affect  others.  This  singular  ob- 
servation is  due  to  Doctor  Wollaston.*  Nothing  can  be  more 
surprising  than  to  see  two  persons,  neither  of  them  deaf,  the  one 
complaining  of  the  penetrating  shrillness  of  a  sound,  while  the 
other  maintains  that  there  is  no  sound  at  all.j-  Few  musical  in- 
struments comprehend  more  than  six  octaves,  and  the  human 
voice  has  only  from  one  to  three,  the  male  voice  being  in  pitch 
an  octave  lower  than  the  female.J 

556.  The  intervals  of  the  diatonic  scale  are  denoted  by  the 
first  seven  letters  of  the  alphabet/A,  B,  C,  D,  E,  F,  G  ;  which 
are  repeated  usually  in  small  letters,  a,  b,  c,  &c.,  in  the  higher 
series. 

A  succession  of  single  musical  sounds  constitutes  melody ;  the 
combination  of  such  sounds,  at  proper  intervals,  forms  chords  ; 
and  a  succession  of  chords  constitutes  harmony.  Two  notes 
produced  by  an  equal  number  of  vibrations  in  a  given  time,  and 
of  course  giving  the  same  sound,  are  said  to  be  in  unison.  The 
relation  between  a  note  and  its  octave  is,  next  after  that  of 
the  unison,  the  most  perfect  in  nature  ;  and  when  the  two  notes 
are  sounded  at  the  same  time,  they  almost  entirely  unite. §  The 
fifth  (Art.  553)  constitutes  the  next  most  perfect  chord,  while  the 
second  and  the  seventh  are  peculiarly  harsh  discords.  By  ex- 
amining the  scale  of  vibrations  in  Art.  553,  we  shall  perceive 
that  the  chords  are  characterized  by  frequent  coincidences  of  vi- 
bralion,  while  in  the  discords  such  coincidences  are  more  rare. 
Thus  in  the  unison,  the  vibrations  are  perfectly  isochronous  ;  in 
the  octave  the  two  coincide  at  the  end  of  every  vibration  of  the 
longer  string,  the  shorter  meanwhile  performing  just  two  vibra- 
tions ;  and  in  the  fifth,  they  coincide  at  the  end  of  every  two  vi- 
brations of  the  longer  string,  the  shorter  vibrating  three  times 
in  the  same  period.  But  in  the  second,  the  longer  and  shorter 
vibrations  can  coincide  only  after  eight  of  the  longer  and  nine 
of  the  shorter,  and  in  the  seventh,  only  after  eight  of  the  longer 
and  fifteen  of  the  shorter.  Hence  the  concord  is  more  perfect  as 
the  common  period  is  shorter. || 

Musical  intervals,  therefore,  are  divided  into  chords  and  dis- 
cords. The  octave,  the  major  fifth,  the  major  and  minor  thirds, 
the  major  and  minor  sixths,  are  concords,  and  are  pleasing  in 
themselves.  The  seconds,  the  sevenths,  the  minor  fifths  and 
major  fourths,  are  discords.  The  chord  consisting  of  the  funda- 
mental note  with  its  third  and  fifth,  and  called  the  harmonic 

*  Phil.  Transac.,  1820.  t  Herschel.  J  Arnott,  El.  Phys.  I,  481. 

§  Ed.  Eilcyc.,  Art.  Music.          ||  Young's  N.  Phil.  I,  391. 


ACOUSTICS.  377 

triad,  forms  the  most  perfect  harmony,  and  contains  the  constitu- 
ent parts  of  the  most  simple  and  natural  melodies.* 

Discords,  however,  are  employed  in  musical  composition;  but 
their  use  is  limited  by  special  rules.  Their  use  does  not  consist 
in  the  excess  or  defect  of  intervals,  which  when  false  produce 
jargon,  not  music;  but  in  the  warrantable  and  artful  use  of  such 
combinations  as,  though  too  disagreeable  for  the  ear  to  dwell 
upon,  or  to  finish  a  musical  period,  yet  so  necessary  are  they  to 
modern  counterpoint,  and  modern  ears,  that  harmony  without 
their  relief,  would  satiate,  and  lose  many  of  its  pleasing  efFects.f 

557.  When  a  long  string  is  made  to  vibrate,  there  are  heard 
not  only  the  note  belonging  to  the  whole  length  of  the  string, 
but  also  more  feebly  the  subordinate  notes  belonging  to  its  half, 
its  third,  its  fourth,  &c.,  thus  giving  to  single  sounds  the  effect  of 
harmony.     Hence  such  subordinate  sounds  are  called,  with  re- 
spect to  the  principal  sound,  its  harmonics.    Often  the  subordinate 
sounds  swell  with  such  force  as  to  overpower  for  a  time  the  fun- 
damental note  ;  and  then  if  the  string  be  carefully  examined,  it 
will  be  found  to  be  vibrating,  not  as  a  whole,  but  in  two,  three, 
or  four  distinct  portions,  with  points  of  rest  between  them.J  The 
sounds  thus  belonging  to  a  single  string,  and  produced  by  its  spon- 
taneous division  into  different  numbers  of  equal  parts,  constitute, 
when  heard  together  or  in  succession,  the  simple  music  of  Nature 
herself.     It  is  produced  in  the  most  perfect  manner  by  the  ka- 
lian Harp. 

558.  Hence  arises  what  is  denominated  the  sympathy  of  sounds. 
If  two  strings  equally  stretched,  and  in  all  other  respects  similar, 
but  one  only  half,  one  third,  or  some  other  aliquot  part  of  the 
length  of  the  other,  be  placed  side  by  side,  and  the  shorter  be 
struck  or  sounded,  the  vibration  will  be  communicated  to  the 
longer  by  the  intervention  of  the  air,  which  will  thus  at  once  be 
thrown  into  a  mode  of  vibration  in  which  the  whole  length  is 
divided  into  segments,  each  equal  to  the  shorter  string.     Here 
the  vibrations  imparted  to  the  string  that  is  struck,  are  communi- 
cated to  the  aerial  pulsations,  which  will  impress  on  any  body 
capable  of  vibrating  in  their  own  time,  an  actual  vibratory  mo- 
tion ;  and  if  a  body  is  susceptible  of  a  number  of  modes  of  vi- 
bration performed  in  different  times,  that  mode  only  will  be  ex- 
cited which  is  synchronous  with  the  aerial  pulsations.     All  other 
motions,  though  they  may  be  excited  for  a  moment  by  one  pul- 
sation, will  be  extinguished  by  a  subsequent  one.     Hence,  if  two 
strings  have  any  mode  of  vibration  in  common,  that  mode  may 
be  excited  by  sympathy  in  either  of  them  when  the  other  is  sound- 

*  Young.  t   Burney. 

t  Arnott,  El.  Phys.  I,  478  ;  Yoaig's  Nat.  PhiL  I,  382 ;   Hauy's  Nat.  Phil.  I,  316 
48 


378  NATURAL   PHILOSOPHY. 

ed,  and  that  only.  For  example,  if  the  length  of  one  string  is 
to  that  of  the  other  as  2  :  3,  and  if  either  be  set  vibrating,  the 
mode  of  vibration,  corresponding  to  a  division  of  the  former  into 
two,  and  of  the  latter  into  three  segments,  will,  if  it  exists  in  the 
one,  be  communicated  by  sympathy  to  the  other.  In  the  vibra- 
tions of  strings,  which,  from  their  small  surface,  can  receive 
nothing  but  a  trifling  impulse  from  the  air,  the  sounds  and  mo- 
tions excited  by  this  sort  of  sympathetic  communication  are  fee- 
ble ;  but  in  vibrating  bodies  which  present  a  large  surface,  they 
become  very  great.  It  is  a  pretty  well  authenticated  feat,  per- 
formed by  persons  of  a  clear  and  powerful  voice,  to  break  a 
drinking  glass  by  singing  its  proper  fundamental  note  close  to  it. 
Looking-glasses  also  are  said  to  have  been  occasionally  broken 
by  music,  the  excursions  of  their  molecules  in  the  vibrations  into 
which  they  are  thrown  being  so  great  as  to  strain  them  beyond 
the  limits  of  their  cohesion.* 

559.  The  theory  of  Musical  Instruments  will  be  readily  under- 
stood from  the  principles  already  explained.  It  will  be  seen  that 
they  all  owe  their  power  of  producing  musical  sounds  to  their 
susceptibility  of  vibrations ;  that  the  force  or  loudness  of  the 
sounds  they  afford  depends  on  the  length  of  the  vibrations,  and 
the  graveness  or  acuteness  of  the  sound,  in  other  words  the  pitch, 
on  their  slowness  or  frequency  ;  and  that  their  chords  depend,  in 
general,  upon  frequency  of  coincidence  in  the  vibrations  that  afford 
the  several  sounds  of  the  concord. 

The  nature  of  stringed  instruments  may  be  learned  from  the 
violin.  Here  the  strings  are  of  the  same  length,  but  differ  in 
weight  and  tension ;  those  designed  to  afford  the  lower  notes 
being  heavier  and  less  strained,  and  those  for  the  higher  notes 
being  lighter  and  more  tense.  The  lengths,  moreover,  are  alter- 
ed by  applying  the  fingers.  The  several  strings  are  usually  so 
adjusted  to  each  other,  that  is,  so  tuned,  that  any  two  contiguous 
strings  make  a  fifth.  Hence  the  fourth  or  highest  stop  on  one 
string  brings  it  into  unison  with  the  string  above  ;  and  the  third 
stop  on  any  string  forms  an  octave  with  the  open  string  next  be- 
low. On  account  of  this  power  of  altering  the  effective  lengths 
of  the  strings  at  pleasure,  of  developing  the  harmonic  sounds  by 
a  skilful  application  of  the  fingers,  and  of  varying  constantly 
the  degrees  of  fulness  or  force  in  each  sound  by  a  dexterous  use 
of  the  bow,  the  violin  becomes,  in  the  hands  of  an  accomplished 
performer,  an  instrument  of  great  power  and  compass,  while  it 
is  capable  of  greater  variety  than  any  other  musical  instrument. 

The  Jlute  affords  an  example  of  wind  instruments.  Here  the 
vibrating  body  is  a  column  of  air,  to  which  different  lengths  are 
given  by  means  of  the  stops  which  are  opened  and  closed  by  the 

*  Herschel  on  Sound. 


ACOUSTICS.  379 

fingers.  The  rapidity  of  the  vibrations,  and  consequently  the 
pitch,  is  also  changed  a  whole  octave  by  the  management  of  the 
breath. 

560.  In  mixed  wind  instruments,  the  vibrations  or  alternations 
of  solid  bodies,  are  made  to  cooperate  with  the  vibrations  of  a 
given  portion  of  air.  Thus,  in  the  trumpet,  and  in  horns  of  va- 
rious kinds,  the  force  of  inflation,  and  perhaps  the  degree  of  ten- 
sion of  the  lips,  determines  the  number  of  parts  into  which  the 
tube  is  divided,  and  the  harmonic  which  is  produced.  The  haut- 
boy and  clarionet  have  mouth-pieces  of  different  forms,  made  of 
reeds  or  canes ;  and  the  reed  pipes  of  an  organ,  of  various  con- 
structions, are  furnished  with  an  elastic  plate  of  metal,  which 
vibrates  in  unison  with  the  column  of  air  which  they  contain. 
An  organ  generally  consists  of  a  number  of  different  series  of 
pipes,  so  arranged,  that  by  means  of  registers,  the  air  proceeding 
from  the  bellows  may  be  admitted  to  supply  each  series,  or  may 
be  excluded  from  it  at  pleasure  ;  and  a  valve  is  opened  when  the 
proper  key  is  touched,  which  causes  all  the  pipes  belonging  to 
the  note  in  those  series  of  which  the  registers  are  open,  to  sound 
at  once.  These  pipes  are  not  only  such  as  are  in  unison,  but 
frequently  also  one  or  more  octaves  above  and  below  the  princi- 
pal note,  and  sometimes  also  twelfths  and  seventeenths,  imitating 
ihe  series  of  natural  harmonics.* 

For  the  further  elucidation  of  this  interesting  subject,  we  are 
compelled  to  refer  to  more  extensive  treatises,  as  Smith's  Har- 
monics, Herschel  on  Sound,  and  particularly,  the  late  work  of 
Professor  Pierce  on  Sound. 

*  Young's  Lectures,  I,  402. 


380  NATURAL  PHILOSOPHY. 


PART    VI. ELECTRICITY. 


561.  ELECTRICITY  is  a  term  derived  from  ^SXT^  the  Greek 
word  for  amber,*  that  being  the  substance  in  which  a  property 
of  the  agent  now  denominated  Electricity  was  first  observed. 

The  ancient  Greek  philosophers  were  acquainted  with  the  fact 
that  amber,  when  rubbed,  acquires  the  property  of  attracting 
light  bodies  ;  hence  the  effect  was  denominated  electrical ;  and 
in  later  times,  the  term  electricity  has  been  used  to  denote  both 
the  unknown  cause  of  electrical  phenomena,  and  the  science 
which  treats  of  electrical  phenomena  and  their  causes. 

The  science  of  electricity  is  hardly  more  remarkable  on  ac- 
count of  its  surprising  and  beautiful  phenomena,  than  it  is  cu- 
rious in  its  history.  The  first  observation  recorded  of  it  was 
made  by  Thales  of  Miletus,f  who  ascribed  it  to  the  functions  of 
some  hidden  animal.J  Theophrastus,§  the  natural  historian, 
mentions  a  stone  called  lyncurium,  (supposed  to  be  the  tourmalin 
of  modern  mineralogists,)  possessing  the  property  of  attraction 
as  well  as  amber.  He  observes  that  it  is  said  not  only  to  attract 
straws  and  small  pieces  of  sticks,  but  even  copper  and  iron,  if 
they  be  finely  divided.||  This  is  nearly  the  amount  of  what  was 
known  of  electricity  by  the  ancients ;  nor,  so  far  as  appears, 
was  there  a  single  important  fact  added  to  the  science  for  the 
period  of  nineteen  centuries. 

562.  In  the  year  1600,  Dr.  Gilbert,  an  English  philosopher, 
published  a  work  on  Magnetism,  comprising  also  many  observa- 
tions on  Electricity.     He  knew  nothing  more  of  this  agent,  how- 
ever, than  as  a  power  of  attraction.     Little  was  added  to  the 
knowledge  of  Gilbert  on  this  subject  until  the  latter  part  of  the 
same  century,  when,  after  the  establishment  of  the  Royal  Society 

*  Amber  is  a  resinous  substance  having  the  appearance  of  indurated  honey.  It 
sometimes  naturally  exhibits  the  shape  of  water-worn  pebbles.  When  heated,  it  ex- 
hales a  highly  agreeable  odor.  From  its  scarcity  it  bears  a  high  price.  Much  of  the 
amber  found  in  the  market  is  brought  from  Prussia,  where  it  is  found  in  mines,  or 
loosely  scattered  along  the  sea  coast ;  and  it  is  found  in  other  countries,  imbedded  in 
a  peculiar  kind  of  sand  and  gravel. 

t  Sometimes  styled  the  "  father  of  Grecian  philosophy."  Flourished  600  years 
before  the  Christian  era. 

t  Priestley's  History  of  Electricity,  p.  1. 

§  Lived  at  Athens,  300  years  B.  C. 

||  To  \vyKo6fiov  c\xct  yap  S,avcp  TO  JjAtKTpov.  ol  ii  <j,aciv  ov  rfvov  t<ip<prj  Kai  ftXor,  JAAu  coi 
KOI  al&ripov,  lav  %  \titr6s .  Siimcp  KOI  AioicAijj  IXtye. — TheophrastUS  irepl  rZv  \lduv. 


ELECTRICITY.  •  381 

of  London,  and  of  the  Academy  of  Sciences  at  Paris,  philosoph- 
ical experiments  began  to  be  prosecuted  with  a  zeal  before  un- 
known. Boyle*  discovered  a  number  of  interesting  facts  in  elec- 
tricity, and  Otto  Guericke-\  constructed  the  first  electrical  ma- 
chine, using  a  globe  of  sulphur,  instead  of  the  glass  cylinder,  at 
present  employed. 

But  the  first  sixty  years  of  the  eighteenth  century,  may  be  re- 
membered as  the  period  when  the  greatest  discoveries  in  electri- 
city were  made.  Grey,\  in  England,  Du  Fay,§  in  France,  and 
Franklin,\\  in  America,  are  the  names  most  distinguished  in  the 
history  of  this  period.  Each  of  these  individuals  made  numer- 
ous and  important  discoveries  ;  and  the  last  two  severally  pro- 
posed hypotheses  to  account  for  the  phenomena  of  electricity, 
hypotheses  which  have  ever  since  divided  the  opinions  of  elec- 
tricians. 

For  the  sake  of  convenience,  the  term  electric  fluid  is  employed, 
without,  however,  implying  any  thing  more  than  the  unknown 
cause  of  electrical  phenomena,  whatever  that  cause  may  be. 

*  Honorable  Robert  Boyle,  an   English  philosopher,  lived   in  the  reign  of  Charles 
the  Second,  and  flourished,  about  the  year  1670.     He  was  one  of  the  founders  of  the 
Royal  Society  of  London,  and  was  a  very  zealous  and  diligent  experimenter,  and  dis- 
tinguished for  his  virtues  and  piety.     Though  the  facts  discovered  by  Boyle  were  val- 
uable contributions  to  the  science,  yet  it  may  serve  to  show  the  absurd  notions  which 
prevailed  at  that  time  on  points  of  theory,  to  recite  his  views  of  electrical  attraction. 
He  supposed  that  an  excited  body  emitted  a  glutinous  effluvium,  which  laid  hold  of 
email  bodies  in  its  way,  and,  in  its  return  to  the  body  which  emitted  it,  carried  them 
back  with  it— Priestley's  Hist.  Elec.  p.  7. 

t  Otto  Guerickc,  of  Magdeburg  in  Germany,  better  known  as  the  inventor  of  the 
air-pump.  He  was  contemporary  with  Boyle,  and  united  an  inventive  talent  with  a 
taste  for  philosophical  experiments.  His  electrical  machine  consisted  of  a  globe  of 
E-ulphur,  made  by  melting  that  substance  in  a  hollow  globe  of  glass,  and  then  remov- 
ing the  glass  by  breaking  it.  This  globe  he  mounted  upon  an  axis,  and  whirled  it  in 
a  wooden  frame,  rubbing  it  at  the  same  time  with  his  hand.  Guericke  first'  observed 
the  electric  spark. 

*  Stephen  Grey,  a  pensioner  of  the  British  government — flourished  about  the  year 
1730 — made  numerous  discoveries.,  the  most  important  of  which  was  the  division  of 
bodies  into  conductors  and  non-conductors. 

§  Du  Fay  was  a  member  of  the  Academy  of  Sciences  at  Paris — flourished  about 
the  year  1733 — he  discovered,  among  other  things,  the  influence  of  moisture  upon  the 
conducting  power  of  bodies — the  fact  that  electrified  attract  unelectrified  bodies — and 
the  two  different  kinds  of  electricity,  the  vitreous  and  resinous,  or  positive  and  neg- 
ative. 

H  Dr.  Franklin  commenced  his  labors  in  electricity  in  1747.  The  results  of  his 
experiments  and  observations  were  communicated  in  several  letters  addressed  to 
Peter  Collinson,  Esq.,  of  London,  Fellow  of  the  Royal  Society,  written  at  different 
times  from  1747  to  1754.  "  Nothing,"  says  Dr.  Priestley,  (Hist.  Elec.  p.  159,)  "  was 
ever  written  upon  the  subject  of  electricity,  which  was  more  generally  read  and  ad- 
mired in  all  parts  of  Europe,  than  these  letters.  There  is  hardly  any  European  lan- 
guage into  which  they  have  not  been  translated  ;  and,  as  if  this  was  not  sufficient  to 
make  them  properly  known,  a  translation  of  them  has  lately  been  made  into  Latin. 
It  is  not  easy  to  say,  whether  we  are  most  pleased  with  the  simplicity  and  perspicuity 
with  which  these  letters  are  written,  the  modesty  with  which  the  author  proposea 
every  hypothesis  of  his  own,  or  the  noble  frankness  with  which  he  relates  his  mis- 
takes, when  they  were  corrected  by  subsequent  experiments." 


NATURAL   PHILOSOPHY. 


CHAPTER  I. 

OF  THE  GENE/IAL  PRINCIPLES  OF  THE  SCIENCE. 

563.  THE  most  general  effect  by  which  the  presence  of  elec- 
tricity is  manifested,  is  attraction.     Thus,  when  a  glass  tube  is 
rubbed  with  a  dry  silk  or  woolen  cloth,  it  acquires  the  property 
of  attracting  light  bodies,  as  cotton,  feathers,  &e.     When  by  any 
process  a  body  is  made  to  give  signs  of  electricity,  it  is  said  to  be 
excited.     When  a  body  receives  the  electric  fluid  from  an  excited 
body,  it  is  said  to  be  electrified.     Since  there  is  found  to  be  a 
great  difference  in  bodies  in  regard  to  the  power  of  transmitting 
electricity,  all  bodies  are  divided  into  two  classes,  CONDUCTORS 
and  NON-CONDUCTORS.     Conductors  are  bodies  through  which  the 
electric  fluid  passes  readily ;  non-conductors  atre  bodies  through 
which  the  electric  fluid  either  does  not  pass  at  all,  or  but  very 
slowly.     The  latter  bodies  are  also  denominated  electrics,  because 
it  is  by  the  friction  of  bodies  of  this  class  that  electricity  is  usually 
excited.     An  electrified  body  is  said  to  be  insulated,  when  its 
connection  with  other  bodies  is  formed  by  means  of  non-conduc- 
tors, so  that  its  electricity  is  prevented  from  escaping.     Instru- 
ments employed  to  detect  the  presence  of  electricity  are  denom- 
inated electroscopes ;  such  as  are  employed  to  estimate  its  com- 
parative quantity,  are    called  electrometers.      This   distinction, 
however,  is  neglected  by  some  writers,  and,  to  avoid  the  unneces- 
sary multiplication  of  terms,  it  will  be  neglected  in  the  present 
treatise,  instruments  of  either  kind  being  called  electrometers. 

564.  The  Pendulum  Electrometer,  is   fbrmed  by      Fig.  211. 
suspending  some  light  conducting  substance  by  a 
non-conducting  thread.     Thus,  a  small  ball  of  the 

pith  of  elder  hung  by  a  silk  thread,  constitutes  a  very 
convenient  instrument  for  detecting  the  presence  and 
examining  the  kind  of  electricity.  Fig.  211  repre- 
sents a  pendulum  electrometer,  consisting  of  a  glass 
rod  fixed  in  a  stand,  and  bent  at  the  top  so  as  to  form 
a  hook.  From  this  hook  hangs  a  thread  of  raw 
silk,  to  the  bottom  of  which  is  attached  a  small  pith 
ball,  made  smooth  and  round,  and  weighing  only  a 
small  part  of  a  grain.  The  attenuated  thread  of 
silk,  unwound  from  the  ball  of  the  silk- worm,  forms 
a  very  delicate  insulator  ;  but  for  ordinary  purposes, 
a  common  thread  of  silk  may  be  untwisted,  and  a 
single  filament  taken  for  the  suspending  thread.  For  the  pur- 


ELECTRICITY. 


poses  of  the  learner,  it  may  even  be  sufficient  to  suspend,  by  a 
thread  of  silk,  a  ball  of  cork,  or  a  lock  of  cotton,  or  a  feather. 


Fig.  212. 


565.  The  Gold  Leaf  Electrometer,  represented  in 
Fig.  212,  consists  of  two  strips  of  gold  leaf,  suspended 
from  the  metallic  cover  of  a  small  glass  cylinder.  By 
this  arrangement,  the  pieces  of  gold  leaf  are  insulated, 
they  are  protected  from  agitation  by  the  air,  and  elec- 
tricity is  easily  conveyed  to  them  by  bringing  an  elec- 
trified body  into  contact  with  the  cover.  The  ap- 
proach of  an  electrified  body  causes  the  leaves  to  sep- 
arate, or  when  previously  separated,  to  collapse,  ac- 
cording to  principles  to  be  explained  presently. 


566.  Coulomb's  Electrometer,  Fig.  213,  is  an 
apparatus  of  still  greater  delicacy  and  perfec- 
tion than  either  of  the  preceding  instruments. 
It  consists  of  a  cylindrical  glass  vessel,  having 
also  a  lid  of  glass,  in  the  center  of  which  a 
small  hole  is  drilled.     Through  this  hole  passes 
an  untwisted  raw  silk  thread  four  inches  long, 
and  fixed  at  the  top  to  a  micrometer,  by  means 
of  which  it  may  be  turned  round  any  number 
of  degrees  at  pleasure.     To  the  silk  thread  is 
attached  a  very  fine  thread  of  lac,  H,  having 
at  each  extremity  a  small  pith  ball.     This  lac 

needle  with  its  knobs  weighs  only  one  fourth  of  a  grain.  A 
small  hole  is  drilled  in  the  side  of  a  vessel  at  A,  through  which 
passes  a  fine  wire,  terminated  at  both  extremities  by  a  knob. 
When  an  excited  body  is  placed  in  contact  with  the  knob  at  A, 
the  knob  at  the  other  extremity  will  acquire  the  same  electricity 
as  the  excited  body.  This  electricity  it  will  communicate  to 
the  knob  of  the  lac  needle,  suspended  by  the  silk  thread,  which 
was  previously  almost  in  contact,  and  the  two  knobs  will  repel 
each  other.  The  movable  knob  attached  by  the  silk  thread, 
will  separate  from  the  other,  and  the  quantity  of  electricity  will 
be  proportional  to  the  distance  to  which  it  recedes.* 

By  the  aid  of  the  foregoing  instruments,  or  even  by  means  of 
the  pendulum  electrometer  alone,  we  may  ascertain  the  follow- 
ing LEADING  FACTS,  which  are  so  many  fundamental  truths,  in  the 
science  of  electricity. 

567.  PROP.  I.  Electricity    is   produced  by   the  friction   of  all 
bodies. 

Although  friction  is  the  most  common,  and  by  far  the  most  ex-- 
tensive means  of  exciting  bodies,  yet  it  is  not  the  only  means. 
Electricity  is  manifested  during  the  changes  of  state  in  bodies, 


*  Thomson's  Outlines  of  Heat  ar.I  Electricity,  p.  374. 


384  NATURAL   PHILOSOPHY. 

such  as  liquefaction  and  congelation,  evaporation  and  condensa- 
tion. Some  bodies  even  are  excited  by  mere  pressure ;  others  by 
the  contact  or  separation  of  different  surfaces.  Most  chemical 
combinations  and  decompositions  are  also  attended  by  the  evolu- 
tion of  electricity,  which  manifests  its  presence  to  delicate  elec- 
trometers. 

If  we  rub  a  piece  of  amber,  sealing  wax,  or  any  other  resin 
ous  substance,  on  dry  woolen  cloth,  or  fur,  or  silk,  and  bring  it 
toward  an  electrometer,  it  will  give  signs  of  electricity.  A  glass 
tube  may  be  excited  in  a  similar  manner.  Moreover,  if  we 
bring  the  excited  .tube  near  the  face,  it  imparts  a  sensation  re- 
sembling that  produced  by  a  cobweb.  If  the  tube  is  strongly 
excited,  it  will  afford  a  spark  to  the  knuckle,  accompanied  by  a 
snapping  noise.  A  sheet  of  white  paper,  first  dried  by  the  fire, 
and  then  laid  on  a  table  and  rubbed  with  India  rubber,  will  be- 
come so  highly  excited  as  to  adhere  to  the  wall  of  the  room,  or 
any  other  surface  to  which,  it  is  applied.  Indeed,  friction  is  so 
constantly  attended  by  electricity,  that  in  favorable  weather  the 
fluid  is  abundantly  indicated  on  brushing  our  clothes,  which  thus 
are  made  to  attract  the  light  downy  particles  that  are  floating 
in  the  air. 

568.  Our  proposition  asserts,  that  electricity  is  produced  by  the 
friction  of  all  bodies,  whereas,  if  we  hold  in  the  hand  a  metallic 
substance,  a  plate  of  brass  or  iron,  for  example,  and  subject  it  to 
friction,  we  shall  not  discover  the  least  sign  of  electrical  excite- 
ment.    In  such  cases,  however,  the  electricity  is  prevented  from 
accumulating  in  consequence  of  the  substance  being  a  good  con- 
ductor, and  thus  conveying  the  fluid  to  the  hand,  which  is  an- 
other good  conductor,  by  which  means  it  is  lost  as  fast  as  it  is 
excited.     But  if  we  insulate  a  metallic  body,  or  any  other  con- 
ducting substance,  then,  on  being  rubbed,  it  gives  signs  of  elec- 
tricity, like  electrics. 

Liquids  and  gases,  by  friction  against  solid  bodies,  excite  elec- 
tricity. Thus,  quicksilver  rapidly  agitated  in  a  glass  tube  elec- 
trifies it,  and  the  blast  of  a  bellows  against  the  projecting  knob 
of  Coulomb's  electrometer,  (see  Fig.  213,)  puts  the  needle  in 
motion.  Even  a  slight  puff  with  the  mouth,  directed  upon  the 
knob,  will  produce  a  sensible  degree  of  excitation. 

569.  PROP.  II.   The  electricity  which  is  excited  from  GLASS  and  a 
numerous  class  of  bodies,  exhibits  different  properties  from  that 
which  is  excited  from  AMBER,  or  seating  wax,  and  a  class  of  bodies 
equally  numerous  with  the  other. 

'The  kind  of  fluid  excited  from  glass  and  analogous  bodies  is 
called  vitreous,  and  that  from  amber  and  analogous  bodies,  resin- 
ous electricity.  The  term  positive  is  also  used  instead  of  vitre- 
ous, and  negative  instead  of  resinous. 


ELECTRICITY.  385 

In  order  to  understand  the  applications  of  the  preceding  terms, 
vitreous  and  resinous,  positive  and  negative,  it  is  necessary  to 
know  something  of  the  two  hypotheses  upon  which  these  terms 
are  respectively  founded.  The  first  hypothesis  is  that  proposed 
by  Du  Fay.*  It  ascribes  all  electrical  phenomena  to  the  agency 
of  two  fluids,  specifically  different  from  each  other,  and  perva- 
ding all  bodies.  In  unelectrified  bodies,  these  two  fluids  exist 
in  combination,  and  exactly  neutralize  each  other.  By  the  sepa- 
ration of  the  two  fluids  it  is  that  bodies  are  electrified  ;  and  it  is 
by  the  reunion  of  the  two  fluids,  that  the  electricity  is  discharged, 
or  bodies  cease  to  be  excited.  The  second  hypothesis  was  pro- 
posed by  Dr.  Franklin.  It  ascribes  all  electrical  phenomena  to 
the  agency  of  one  fluid,  which,  as  in  the  other  case,  is  supposed 
to  pervade  all  bodies,  being  naturally  in  a  state  of  equilibrium. 
It  is  only  when  this  equilibrium  is  destroyed  that  bodies  become 
electrified,  and  it  is  by  the  restoration  of  the  equilibrium  that  the 
electricity  is  discharged,  or  bodies  cease  to  be  excited.  But  a 
body  is  electrified  when  it  has  either  more  or  less  of  the  fluid 
than  its  natural  share  ;  in  the  former  case  it  is  positively,  in  the 
latter  case  negatively,  electrified  ;  positive  electricity,  therefore, 
implies  ^a  redundancy,  and  negative  electricity,  a  deficiency  of 
the  fluid. 

This  much  being  sufficient  for  the  understanding  of  the  terms, 
and  of  the  general  principles  of  these  two  celebrated  hypotheses, 
we  shall  postpone  all  discussions  respecting  them,  until  the  learn- 
er has  become  acquainted  with  a  sufficient  number  of  electrical 
phenomena,  to  enable  him  to  understand  and  to  judge  of  the  evi- 
dence adduced  in  support  of  each  hypothesis. 

570.  PROP.  III.  Bodies  electrified  in  different  ways  attract,  and  in 
the  same  way  repel  each  other. 

Thus  if  an  insulated  pith  ball,  (Art.  564,)  or  a  lock  of  cotton, 
oe  electrified  by  touching  it  with  an  excited  glass  tube,  it  will 
immediately  recede  from  the  tube,  and  from  all  other  bodies 
which  afford  the  vitreous  electricity,  while  it  will  be  attracted 
by  excited  sealing  wax,  and  by  all  other  bodies  which  afford  the 
resinous  electricity.  If  a  lock  of  fine  long  hair  be  held  at  one 
end,  and  brushed  with  a  dry  brush,  the  separate  hairs  will  be- 
come electrified,  and  will  repel  each  other.  In  like  manner,  two 
insulated  pith  balls,  or  any  other  light  bodies,  will  repel  each 
other  when  they  are  electrified  the  same  way,  and  attract  each 
other  when  they  are  electrified  different  ways. 

Hence  it  is  easy  to  determine  whether  the  electricity  afforded  by 
a  given  body  is  vitreous  or  resinous ;  for,  having  electrified  the 
electrometer  by  excited  glass,  then  all  those  bodies,  which  when 

*  This  is  sometimes  called  the  hypothesis  of  Symmer,  after  an  English  electrician 
of  that  name,  who  matured  and  illustrated  the  principle  first  suggested  by  Du  Fay. 
(See  Phil.  Trans.  1759.) 

49 


rfOb  NATURAL   PHILOSOPHY. 

excited  attract  the  ball,  afford  the  resinous,  while  all  those  which 
repel  the  ball,  afford  the  vitreous  electricity. 

571.  PROP.  IV.   The  two  kinds  of  electricity  are  produced  simul- 
taneously;  the  one  kind  in  the  body  rubbed,  the  other  in  the  rubber. 

For  example,  if  we  rub  a  glass  tube  with  a  silk  or  woolen 
cloth,  the  glass  becomes  positive  and  the  cloth  negative.  The 
foregoing  law  holds  true  universally  ;  but  the  kind  of  electricity 
which  each  substance  acquires,  depends  upon  the  substance 
against  which  it  is  rubbed.  If  we  rub  dry  woolen  cloth  against 
smooth  glass,  it  acquires  the  resinous,  and  the  glass  the  vitreous 
electricity  ;  but  if  we  rub  the  same  cloth  against  rough  glass,  it 
becomes  positively,  while  the  glass  becomes  negatively  electri- 
fied.* The  following  table  contains  a  number  of  electric  sub- 
stances, arranged  in  such  a  way  that  when  they  are  rubbed 
against  each  other,  any  substance  in  the  list  before  another  be- 
comes positively,  and  any  substance  below  it  negatively  electri- 
fied: 

1.  Fur  of  a  Cat,  6.  Paper, 

2.  Smooth  Glass,  7.  Silk, 

3.  Woolen  Cloth,  8.  Lac, 

4.  Feathers,  9.  Rough  Glass, 

5.  Wool,  10.  Sulphur. 

The  fur  of  a  cat,  when  rubbed  against  any  of  the  bodies  in  the 
table,  always  affords  the  vitreous,  and  the  sulphur  always  the 
resinous  electricity.  Feathers  become  negative  when  rubbed 
against  the  fur  of  a  cat,  smooth  glass,  or  woolen  cloth  ;  but  pos- 
itive when  rubbed  against  wool,  paper,  silk,  lac,  rough  glass,  or 
sulphur,  f 

572.  PROP.  V.  Electricity  passes  through  some  bodies  with  the 
greatest  facility ;  through  others  with  the  greatest  apparent  difficulty, 
or  scarcely  at  all ;  and  others  have  a  conducting  power  intermediate 
between  the  two. 

Metals  and  charcoal,  water  and  all  liquids,  (oils  excepted,)  are 
good  conductors.  Melted  wax  and  tallow  are  good  conductors  ; 
but  these  bodies  while  solid  conduct  very  badly.  Glass,  resins, 
gums,  sealing  wax,  silk,  sulphur,  precious  stones,  oxides,  and  all 
gases,  are  non-conductors,  or  at  least  very  bad  conductors.J 
Atmospheric  air  is  a  non-conductor  of  the  highest  class,  when 
perfectly  dry  ;  but  it  becomes  a  conductor  either  when  moist  or 
when  rarefied.  The  electric  fluid  easily  pervades  the  vacuum 

*  The  cloth  should  be  attached  to  a  glass  handle  to  insulate  it. 

t  When  black  stockings  are  worn  over  white,  numerous  sparks  are  frequently  ob- 
served on  pulling  off  the  outer  pair.  The  same  appearances  occur  when  a  silk  gar- 
ment is  worn  over  flannel.  (See  an  interesting  account  of  Symmer's  experiments  on 
this  subject,  in  Priestley's  History  of  Electricity,  p.  267.) 

t  Thomson. 


ELECTRICITY.  387 

of  an  air-pump,  or  of  the  Torricellian  tube,  (Art.  460  ;)  but  these 
are  imperfect  vacuums :  it  is  said  that  electricity  cannot  pass 
through  a  perfect  vacuum.* 

The  conducting  powers  of  most  bodies  are  influenced  by  chan- 
ges of  temperature,  and  also  by  changes  of  form.  Water,  in  its 
natural  state,  is  a  good  conductor  ;  but  its  conducting  power  is 
increased  by  heat  and  diminished  by  cold.  Steam  and  ice  .  are 
each  inferior,  in  conducting  power,  to  pure  water  ;  and  ice  be- 
low the  temperature  of —  13°  Fah.  becomes  an  electric  of  the 
highest  class.  Snow,  when  cold  and  dry,  is  a  bad  conductor. 
During  a  dry  snow  storm  the  air  frequently  becomes  highly  elec- 
trical. 

The  same  body  frequently  exhibits  great  changes  in  conduct- 
ing power  by  changes  of  state  or  chemical  constitution.  Thus, 
green  wood  is  a  conductor,  dry  baked  wood  a  non-conductor ; 
charcoal  a  conductor,  ashes  a  non-conductor. 

573.  Strictly  speaking,  there  is  no  substance  known  that  is 
entirely  impervious  to  electricity  ;  for  the  intensity  of  that  agent 
may  be  so  increased  as  to  force  it,  for  a  greater  or  less  distance, 
through  all  bodies.  Neither  is  there  any  body  in  which  the  con- 
ducting power  is  perfect.  The  following  table  presents  a  cata- 
logue of  bodies  arranged  in  the  order  of  their  conducting  pow- 
ers : 
CONDUCTORS. 

Metals,  the  more  perfect,  or  least  oxidable,  the  better. 

Charcoal,  better  when  prepared  from  hard  wood  and  well 
burned. 

Plumbago. 

Charcoal  in  fine  powder. 

Pure  Water. 

Snow,  better  when  moist,  worse  when  dry. 

Living  Vegetables. 

Living  Animals. 

Flame,  Smoke,  Steam. 

Rarefied  Air. 
NON-CONDUCTORS  OR  ELECTRICS. 

Lac,]  Amber,  Resins. 

Sulphur. 

Wax. 

Fat. 

*  Lib.  Use.  Knowl.,  Art.  Electricity,  p.  5. 

t  Lac,  which  is  placed  at  the  head  of  non-conductors,  is  a  species  of  resin,  sold  by 
the  druggists,  and  is  a  substance  deposited  upon  a  tree  in  India,  by  a  certain  species 
of  insect. — Shell  lac,  the  most  common  form  employed  in  electrical  experiments,  is 
nothing  more  than  lac  in  its  purest  form. — Sealing  wax  is  substituted  for  lac  in  elec- 
trical experiments,  being  made  chiefly  of  that  substance. — Varnishes  also,  which  are 
employed  to  coat  the  surfaces  of  electrical  apparatus,  owe  their  efficacy  to  lac,  of 
which  they  are  chiefly  composed. 


388  NATURAL   PHILOSOPHY. 

Glass,  Gems,  Precious  Stones. 

Silk,  Wool.      . 

Hair,  Feathers. 

Cotton,  Paper. 

Dry  Atmospheric  Air,  and  other  gases. 

Baked  Wood. 

India  Rubber. 

It  is  particularly  important  to  remember  that  Metals,  Water, 
and  all  moist  substances,  Animal  substances,  as  the  human  body, 
and  the  Earth  itself,  are  conductors  ;  while  the  Air,  when  dry, 
and  all  Resinous  and  Vitreous  substances  are  non-conductors. 
These  bodies  are  those  which  are  chiefly  concerned  in  making 
experiments  with  electrical  apparatus. 

574.  PROP.  VI.  Insulation  is  effected  in  various  degrees  of  per- 
fection, according  to  the  state  of  the  atmosphere,  and  the  nature  of 
the  substances  employed  as  insulators. 

If  the  air  were  a  conductor,  it  is  not  easy  to  see  how  the  elec- 
tric fluid  could  be  confined  so  as  to  be  accumulated.  It  is,  more- 
over, only  when  the  air  is  dry  that  it  is  capable  of  insulating 
well ;  hence,  in  damp,  foggy,  and  rainy  weather,  electrical  ap- 
paratus will  not  work  well  unless  the  air  is  dried  artificially  by 
operating  in  a  close  room  highly  heated  by  a  stove.* 

Lac,  drawn  into  fine  threads,  is  the  most  perfect  insulator. 
Compared  with  silk  thread,  such  a  filament  is  ten  times  more  ef- 
fectual in  preventing  the  loss  of  the  fluid.  Fine  silk  thread,  how- 
ever, when  perfectly  dry,  is  among  the  best  insulators,  and  where 
great  delicacy  is  required,  a  single  filament  of  silk  as  it  comes 
from  the  ball  of  the  silk-worm  is  employed.  Its  conducting 
power  is  somewhat  influenced  by  its  color, — black  being  the 
worst,  and  a  gold  yellow  the  best  color  for  insulating.  Glass  is 
much  used  as  an  insulator,  especially  when  great  strength  is  re- 
quired, as  in  supports  to  various  kinds  of  electrical  apparatus. 
Glass,  however,  is  liable  to  acquire  moisture  on  its  surface,  in 
consequence  of  which  its  properties  as  an  insulator  are  materi- 
ally impaired.  This  inconvenience  is  obviated  by  giving  it  a 
thick  coat  of  varnish.  Fine  hair  is  a  good  and  convenient  sub- 
stance in  some  cases  of  insulation. 

In  certain  instances,  conducting  or  uninsulating  threads  are 
required.  Then  fine  silver  wires,  or  linen  threads  first  steeped 
in  a  solution  of  salt  and  dried,  are  used. 

575.  The  sphere  of  communication  is  the  space  within  which 
a  spark  may  pass  from  an  electrified  body,  in  any  direction  from 
it.     It  is  sometimes  called  the  striking  distance.     The  sphere  of 

*  We  have  been  able  to  hold  public  lectures  on  electricity,  illustrated  by  numerous 
experiments,  in  the  most  unfavorable  weather,  by  keeping  the  room  highly  heated  by 
close  stoves. 


ELECTRICITY.  389 

influence  is  the  space  within  which  the  power  of  attraction  of  an 
electrifie'd  body  extends  in  every  way,  beyond  the  sphere  of 
communication.  A  glass  tube  strongly  excited  will  exert  an  in- 
fluence upon  the  gold  leaf  electrometer  at  the  distance  of  ten  or 
even  twenty  feet,  although  a  spark  could  not  pass  from  the  tube 
to  the  cap  of  the  electrometer  at  a  greater  distance  than  a  few 
inches. 

The  electricity  which  a  body  manifests  by  being  brought  near 
to  an  excited  body,  without  receiving  a  spark  from  it,  is  said  to 
be  acquired  by  Induction. 

When  an  insulated  conductor,  unelectrified,  is  brought  into 
the  neighborhood  of  an  insulated  charged  conductor,  its  electri- 
city undergoes  a  new  arrangement.  The  end  of  it  next  to  the 
-excited  conductor,  assumes  a  state  of  electricity  opposite  to  that 
of  the  excited  conductor ;  while  the  farther  extremity  assumes 
the  same  kind  of  electricity.  Suppose  the  excited  conductor  is 
electrified  positively.  The  end  of  the  insulated  conductor  next 
to  it  becomes  negative,  and  the  remoter  end  positive  ;  and  inter- 
mediate between  these  two  points,  there  occurs  a  place  where 
neither  positive  nor  negative  electricity  can  be  perceived.  This 
place  is  called  the  neutral  point. 

The  reason  why  unelectrified  bodies  are  attracted  by  excited 
electrics  is,  that  they  are  put  into  the  opposite  state  by  induction, 
and  then  attracted  upon  the  general  principle  laid  down  in  Prop. 
III.  When  they  come  into  the  sphere  of  communication  of  the 
excited  body,  they  immediately  acquire  the  same  kind  of  elec- 
tricity, and  are  repelled.  If  they  come  into  contact  with  unin- 
sulated bodies,  they  lose  the  electricity  they  have  acquired,  are 
again  put  into  the  opposite  state  by  induction,  again  attracted 
and  again  repelled.  This  process  will  go  on  until  the  electricity 
of  the  insulated  conductor  is  all  conveyed  away.* 

A  body  which  has  been  electrified  by  induction,  returns  to  its 
natural  state  instantaneously  when  the  electrifying  body  is  sud- 
denly withdrawn.  This  is  called  the  return  stroke.  Thunder 
clouds  sometimes  put  objects  beneath  them  under  so  powerful  an 
influence  of  induction,  that  on  the  return  to  the  natural  state  the 
shock  is  so  violent  as  to  destroy  life. 

The  foregoing  general  principles  may  be  verified  with  very 
simple  apparatus,  such  as  pith  balls,  a  glass  tube,  and  a  stick  of 
sealing  wax.  But  the  same  facts  may  be  exhibited  in  a  much 
more  striking  and  impressive  manner  by  the  electrical  machine 
and  its  appendages,  and  our  attention  will  therefore  be  now  turn- 
ed to  the  consideration  of  the  subject  of  electrical  apparatus. 

*  Thomson's  Outlines  of  Heat  an:  Electricity,  p.  362. 


390  NATUKAL   PHILOSOPHY. 

• 

CHAPTER  II. 

OF  ELECTRICAL  APPARATUS. 

576.  THE  object  of  the  electrical  machine  is  to  accumulate 
electricity.  It  is  made  of  several  different  forms,  but  two  of 
these  forms  are  predominant,  which  it  will  be  sufficient  for  our 
present  purpose  to  describe  ;  of  these,  one  is  called  the  Cylinder, 
the  other,  the  Plate  Machine.  The  CYLINDER  MACHINE  is  repre- 
sented in  Fig.  214.  The  principal  parts  belonging  to  it,  are  the 
cylinder,  the  frame,  the  rubber,  and  the  prime  conductor.  The 
cylinder  (A)  is  of  glass,  from  eight  to  twelve  inches  in  diameter, 
and  from  twelve  to  twenty-four  inches  long.  It  should  be  per- 
fectly cylindrical,  otherwise  it  will  not  press  the  cushion  or  rub- 
ber evenly  when  turned.  It  must  be  as  smooth  as  possible,  for 
rough  glass  becomes  a  partial  conductor ;  the  former  only  is 
suitable  for  affording  positive  electricity.  The  cylinder  should 
be  so  mounted  on  the  frame  as  to  revolve  without  waddling,  for 
such  a  motion  would  prevent  its  being  in  uniform  contact  with 
the  rubber.  The  frame  (B,  B)  is  made  of  wood,  which  must  be 
close-grained,  well  seasoned,  and  baked  in  an  oven,  and  finally 
coated  with  varnish,  the  object  of  all  this  preparation  being  to 
Fig.  214 


dimmish  its  conducting  powers,  and  thus  prevent  its  wasting 
the  electricity  of  the  cylinder.  The  rubber  (C)  consists  of  a 
leathern  cushion,  stuffed  with  hair  like  the  padding  of  a  saddle. 
This  is  covered  with  a  black  silk  cloth,  having  a  flap  which  ex- 
tends from  the  cushion  over  the  top  of  the  cylinder  to  the  distance 


ELECTRICITY.  391 

of  an  inch  from  the  points  connected  with  the  prime  conductor, 
to  be  mentioned  presently.  The  rubber  is  coated  with  an 
amalgam,*  made  of  mercury,  zinc,  and  tin,  which  preparation 
has  been  found,  by  experience,  to  produce  a  high  degree  of 
electrical  excitement,  when  subjected  to  the  friction  of  glass. 
The  rubber  is  insulated  by  placing  it  on  a  solid  glass  pillar,  and 
it  is  made  to  fit  closely  to  the  cylinder  by  means  of  a  spring 
worked  by  a  screw. 

The  prime  conductor  (D)  is  usually  a  hollow  brass  cylinder 
with  hemispherical  ends.  It  is  mounted  on  a  solid  glass  pillar, 
with  a  broad  and  heavy  foot,  made  of  wood,  to  keep  it  steady. 
The  cylinder  is  perforated  with  small  holes,  for  the  reception  of 
wires  (c)  with  brass  knobs. 

It  is  important  to  the  construction  of  an  electrical  machine, 
that  the  work  should  be  smooth  and  free  from  points  and  sharp 
edges,  since  these  have  a  tendency  to  dissipate  the  fluid,  as  will 
be  more  fully  understood  hereafter.  For  a  similar  reason,  the 
machine  should  be  kept  free  from  dust,  the  particles  of  which 
act  like  points,  and  dissipate  the  electricity. 

577.  The  PLATE  MACHINE,  (Fig.  215,)  consists  of  a  circular 
plate  of  glass  from  eighteen  to  twenty-four  inches  or  more  in 
Fig.  215. 


*  The  amalgam  recommended  by  Singer,  one  of  the  ablest  practical  electricians, 
is  composed  of  zinc  two  ounces,  of  tin  one  ounce,  and  of  mercury  six  ounces.  The 
zinc  and  tin  may  be  melted  together  in  a  ladle  or  crucible,  and  poured  into  a  mortar 
previously  heated,  to  prevent  the  sudden  congelation  of  the  melted  metals.  As  soon 


392  NATURAL   PHILOSOPHY. 

diameter,  turning  vertically  on  an  axis  that  passes  through  its 
center.  The  frame  is  composed  of  materials  similar  to  those 
which  compose  the  frame  of  the  cylindrical  machine.  This 
machine  is  furnished  with  two  pairs  of  rubbers,  attached  to  the 
top  and  bottom  of  the  plate.  The  prime  conductor  consists  of  a 
brass  cylinder,  proceeding  from  the  center  in  a  line  with  the 
axis,  and  having  two  branches,  which  serve  to  increase  its  sur- 
face, and  at  the  same  time  to  connect  it  with  the  opposite  sides 
of  the  plate,  so  as  to  receive  the  electricity  as  it  is  evolved  from 
each  cushion. 

It  is  not  agreed  which  of  these  two  machines  affords  the  great- 
est quantity  of  electricity  from  the  same  surface  ;  but  the  cylin- 
der is  less  expensive  than  the  plate,  and  less  liable  to  break,  and 
is  more  convenient  for  common  use. 

578.  The  principles  of  the  electrical  machine,  will  be  readily 
comprehended  from  what  has  gone  before.     It  differs  from  the 
glass  tube,  only  in  affording  a  more  convenient  and  effectual 
mode  of  producing  friction.     By  the  friction  of  the  glass  cylinder 
or  plate  against  the  rubber,  electricity  is  evolved,  which  is  im- 
mediately transferred  to  the  prime  conductor,  and  may  be  taken 
from  the  latter  by  the  knuckle,  or  any  other  conducting  substance. 
If  the  glass  and  the  rubber  both  remain  insulated,  the  quantity 
of  electricity  which  they  are  capable  of  affording,  will  soon  be 
exhausted.     Hence,  a  chain  or  wire  is  hung  to  the  rubber  and 
suffered  to  fall  upon  the  table  or  floor,  which,  communicating  as 
it  does  with  the  walls  of  the  building,  and  finally  with  the  earth, 
supplies  an  inexhaustible  quantity  of  the  fluid  to  the  rubber. 
In  cases  where  verj-  great  quantities  of  electricity  are  required, 
a  metallic  communication  may  be  formed  immediately  between 
the  rubber  and  .the  ground. 

579.  In  order  to  indicate  the  degree  of  excitation  in  the  prime 
conductor,  the  Quadrant  Electrometer  is  attached  to  it.  as  is  rep- 
resented at  E  in  Fig.  214.     This  electrometer  is  formed  of  a 
semicircle,  usually  of  ivory,  divided  into  degrees  and  minutes, 
from  0°  to  180°,*  the  graduation  beginning  at  the  bottom  of  the 
arc.     The  index  consists  of  a  straw,  moving  on  the  center  of  the 
disk,  and  carrying,  at  the  other  extremity,  a  small  pith  ball.  The 
perpendicular  support  is  a  pillar  of  brass,  or  some  conducting 

as  they  are  introduced,  they  must  be  rapidly  stirred  with  the  pestle,  during  which 
process  the  mercury  may  be  added,  and  the  stirring  continued,  until  the  amalgam  is 
cold,  when  it  will  be  in  the  form  of  paste  or  fine  powder.  A  little  lard  is  added,  to 
give  the  amalgam  the  proper  consistence  ;  but,  if  when  applied  it  be  warmed  a  little, 
but  a  small  proportion  of  lard  need  be  used.  In  hot  weather,  less  quicksilver  is  to  be 
employed. 

*  Sometimes  the  division  is  carried  only  to  90°.  which  is  all  that  is  necessary 


ELECTRICITY.  393 

substance.  When  this  instrument  is  in  a  perpendicular  position, 
and  not  electrified,  the  index  hangs  by  the  side  of  the  pillar,  per- 
pendicularly to  the  horizon ;  but  when  the  prime  conductor  is 
electrified,  it  imparts  the  same  kind  of  electricity  to  the  index, 
repels  it,  and  causes  it  to  rise  on  the  scale  toward  an  angle  of 
90°,  or  to  a  position  at  right  angles  with  the  pillar.  It  is  obvious 
that  the  index  can  never  rise  higher  than  90°,  since  the  knob 
which  terminates  the  brass  pillar  is  electrified  to  the  same  de- 
gree as  the  prime  conductor,  and  repels  the  index  with  equal 
force.*  Nor  is  the  angle  at  which  the  index  remains  suspended 
to  be  regarded  as  the  true  measure  of  the  repulsive  force.  It 
has  been  demonstrated,  that,  in  order  to  estimate  this  force  truly, 
the  arc  of  the  electrometer  should  be  divided  according  to  a 
scale  of  arcs,  the  tangents  of  which  are  in  arithmetical  progres- 
sion.! 

580.  When  an  electrical  *machine  is  skillfully  fitted  up,  and 
works  well,  on  turning  it,  circles  of  light  surround  the  cylinder 
or  plate,  and  brushes  or  pencils  of  light  emanate  copiously  from 
the  cushion  and  other  parts  of  the  machine.  The  circles  of  light 
consist  of  electric  sparks,  which  discharge  themselves  between 
the  excited  surface  and  the  rubber,  their  passage,  being  so  rapid 
as  to  appear  like  a  continued  line,  like  that  of  a  small  stick  igni- 
ted at  the  end  and  whirled  in  the  air.  The  brushes  of  light  arise 
from  the  facility  with  which  the  fluid  escapes  from  points  or  thin 
edges. 

The  experiments  which  were  previously  performed  on  electri- 
cal attractions  and  repulsions,  (Arts.  567 — 575,)  may  now  be  re- 
peated in  a  much  more  striking  manner,  and  various  other  exper- 
iments added,  which  can  be  shown  only  when  electricity  is  ac- 
cumulated. 


*  Sometimes  when  the  knob  is  so  small  that  the  electricity  escapes  from  it  as  from 
a  point,  it  does  not  repel  the  index  with  the  same  force  as  the  prime  conductor ;  in 
which  case  the  index  rises  above  90°. 

T  Partington's  Manual,  Nat.  Phil.,  II,  157. — As  electrical  machines  are  expensive, 
and  not  always  easily  procured  by  the  private  learner,  it  may  be  useful  to  suggest  a 
mode  of  fitting  up  a  cheap. apparatus.  A  large  tincture  bottle  may  be  procured  of  the 
apothecary,  for  the  cylinder.  A  cover  of  wood  may  be  cemented  to  each  end,  to  the 
center  of  which,  next  to  the  bottom,  is  screwed  a  projecting  knob  for  one  end  of  the 
axis,  while  the  part  of  the  axis  to  which  the  handle  is  attached,  is  screwed  into  the 
center  of  the  cover  of  wood  next  to  the  nozzle.  Thus  prepared,  it  may  be  mounted 
on  such  a  frame  of  hard  dry  wood  as  every  joiner  or  cabinet  maker  can  construct. 
A  tinner  can  make  the  prime  conductor,  and  several  other  appendages  to  be  described 
hereafter.  Junk  bottles  or  long  phials  serve  well  as  insulators.  Ingenious  students 
of  electricity  frequently  amuse  themselves  with  making  machines  of  this  description, 
some  of  which  have  answered  nearly  every  purpose  of  the  most  expensive  kinds  ol 
apparatus. 

A  cement,  for  electrical  purposes,  may  be  made  by  melting  together  five  ounces  of 
resin,  one  ounce  of  beeswax,  one  ounce  of  Spanish  brown,  and  a  teaspoonful  of  plas- 
ter of  Paris,  or  brick  dust. 

50 


394  NATURAL    PHILOSOPHY. 

581.  We  proceed  to  enumerate  a  few  of  the  effects  of  electri- 
city, as  they  are  exhibited  by  the  electrical  machine,  confining 
ourselves  for  the  present  to  those  experiments  which  relate  to  at- 
traction and  repulsion,  and  the  passage  of  the  spark,  reserving 
sucn  as  relate  to  light  and  heat  to  future  sections.  The  follow- 
ing effects  may  be  observed  with  a  machine  of  moderate  pow- 
ers, the  rationale  of  which  the  learner  will  readily  supply  from 
the  propositions  given  in  Art.  567,  &c. 

(1.)  When  the  machine  is  turned,  a  downy  feather,  or  a  lock 
of  cotton  held  in  the  hand  by  a  conducting  thread,*  will  be  strong- 
ly attracted  toward  the  excited  surface. 

(2.)  A  skein  of  thread,  or  lock  of  fine  hair,  looped,  and  sus- 
pended by  the  loop  from  the  prime  conductor,  will  exhibit  strong 
repulsions  between  the  threads  or  hairs.  Lamp-wick,  of  light 
and  spongy  cotton,  furnishes  the  best  threads  for  this  experiment. 

(3.)  The  quadrant  electrometer  (Art.  579)  being  attached  to 
the  prime  conductor,  the  conducting  powers  of  different  sub- 
stances may  be  readily  tried.  Thus,  an  iron  rod  held  in  the 
hand,  and  applied  to  the  prime  conductor,  will  cause  the  index 
of  the  electrometer  to  fall  instantly ;  and  the  same  effect  will 
follow  the  application  of  any  metallic  rod.  A  wooden  rod  of  the 
same  dimensions,  will  cause  the  index  to  descend  more  slowly; 
and  a  glass  rod  will  hardly  move  it  at  all.  ,  These  experiments 
show  that  iron  is  a  perfect,  wood  an  imperfect  conductor,  and 
glass  a  non-conductor.  In  the  same  manner  the  conducting 
powers  of  a  stick  of  sealing  wax,  a  roll  of  silk,  or  cloth,  and  of 
various  other  bodies,  may  be  illustrated. 

(4.)  If  a  pith  ball,  or  leather,  or  any  other  light  body,  held  by 
a  silk  thread,  be  presented  to  the  prime  conductor,  it  will  first  be 
attracted  and  then  repelled,  and  it  cannot  again  be  brought  into 
contact  with  the  electrical  conductor,  until  its  electricity  is  dis- 
charged by  communicating  with  the  finger,  or  some  unelectrified 
conductor. 

(5.)  By  placing  light  bodies  between  an  electrified  conductor 
and  an  uninsulated  body,  they  may  be  made  to  move  with  great 
rapidity  backward  and  forward,  from  one  surface  to  the  other, 
being  alternately  attracted  and  repelled  by  the  electrified  sur- 
face. By  this  means  are  performed  electrical  dances,  the  ring- 
ing of  bells,  and  a  variety  of  interesting  and  amusing  experi- 
ments.f 

(6.)  If  the  rubber  be  insulated  while  the  machine  is  turned, 
the  rubber  and  the  glass  cylinder,  or  plate,  will  be  found  to  be 
in  different  electrical  states ;  an  insulated  body  attracted  by  the 
one  will  be  repelled  by  the  other. 

Bodies  are  electrified  positively  by  connecting  them  with  the 

*  The  conducting  power  of  linen  or  cotton  threads  is  improved  by  moistening 
them  with  the  breath. 

t  See  Singer's  Elements  of  Electricity,  for  a  good  selection  of  these  experiment* 


ELECTRICITY.  395 

glass,  by  means  of  the  prime  conductor,  and  negatively  by  con- 
necting them  with  the  rubber,  the  latter  being  insulated,  and  the 
prime  conductor  uninsulated. 

(7.)  An  electrified  body  frequently  exhibits  a  tendency  to  sep- 
arate into  minute  parts,  these  parts  being  endued  with  the  power 
of  mutual  repulsion.  Thus,  a  lock  of  cotton,  when  electrified, 
is  separated  into  its  minutest  fibres.  Melted  sealing  wax,  when 
attached  by  a  wire  to  the  prime  conductor,  is  divided  into  fila- 
ments so  small  as  to  resemble  red  wool.  Water,  dropping  from 
a  capillary  syphon  tube,  on  being  electrified,  is  made  to  run  out 
in  a  great  number  of  exceedingly  fine  streams.  Water,  spouting 
from  an  air  fountain,  (Art.  458,)  is  divided  into  a  number  of  rays, 
presenting  the  appearance  of  a  brush. 

(8.)  A  portion  of  electrified  air,  in  consequence  of  the  mutual 
repulsion  between  its  particles,  expands,  and  when  at  liberty  to 
escape,  becomes  rarefied.  Thus,  a  current  of  air  may  be  set  in 
motion  from  an  electrified  point,  or  small  ball,  or  be  made  to  is- 
sue from  the  neck  of  a  bottle. 

Such  are  some  of  the  leading  experiments  which  may  be  per- 
formed with  the  common  electrical  machines,  in  addition  to  those 
which  are  connected  with  light  and  heat,  to  be  more  particular- 
ly described  hereafter. 

TORSION  BALANCE. 

*  582.  The  instrument  called  the  Torsion  Balance,  invented  by 
Coulomb,*  exceeds  all  others  in  delicacy  and  the  power  of  meas- 
uring small  forces  ;  and  in  the  skillful  hands  of  the  inventor,  it 
furnished  the  means  of  very  refined  investigations  into  the  most 
hidden  laws  of  electricity.  The  same  instrument  was  also  ap- 
plied to  similar  researches  in  several  other  branches  of  physics, 
affording  in  each  case  an  example  of  the  most  refined  experi- 
mental analysis. 

The  force  employed  to  estimate  any  given  power  of  electric 
attraction,  is  the  force  of  torsion  ;  that  is,  the  effort  made  by  a 
twisted  thread  or  wire  to  untwist  itself.  Since  the  thread  may 
be  small  to  any  extent,  and  may  be  of  any  length,  (and  the  force 
of  torsion  is  found  to  be  inversely  as  the  length,  and  directly  as 
the  fourth  power  of  the  thickness, f)  the  degrees  by  which  this 
force  is  increased  as  the  thread  is  turned,  may  differ  from  each 
other  by  the  smallest  conceivable  quantity,  and  yet  be  separated 
by  spaces  far  enough  asunder  to  be  susceptible  of  being  measured 

*  Charles  Augustus  Coulomb,  was  a  very  distinguished  member  of  the  French  Aca- 
demy, and  remarkable  for  his  assiduity  and  precision  in  experimental  researches.  He 
flourished  during  the  latter  part  of  the  last  century.  His  experiments  on  electricity, 
magnetism,  friction,  and  the  resistance  of  fluids,  are  among  the  finest  in  natural  phi- 
losophy. 

t  Biot,  Precis  El.  tome  I,  339. 


396  NATURAL   PHILOSOPHY. 

with  the  utmost  precision  ;  and  thus  any  force,  as  that  of  elec 
trical  attraction,  required  to  hold  the  successive  degrees  of  the 
force  of  torsion  in  equilibrium,  may  be  exactly  ascertained.  If, 
by  a  fine  thread,  (which  may  be  either  the  smallest  filament  of 
silk,  or  the  finest  silver  wire,)  we  suspend  a  horizontal  needle, 
as  in  the  electrometer  represented  in  Fig.  213,  the  least  conceiv- 
able force  applied  at  the  extremities  of  the  needle,  will  put  it  in 
motion.  A  lever  an  inch  long,  suspended  by  a  fibre  of  silk  four 
inches  in  length,  requires  a  force  only  the  sixty  thousandth  part 
of  a  grain,  to  twist  it  three  hundred  and  sixty  degrees. 

583.  The  construction  of  the  instrument  is  as  Fig.  216. 
follows.  In  order  to  guard  the  suspended  needle 
from  the  agitations  of  the  air,  it  is  protected  by  a 
glass  cylinder,  AB,  having  a  movable  lid,  C,  from 
the  center  of  which  rises  a  smaller  glass  cylinder, 
D,  which  covers  the  suspending  thread ;  this  latter 
cylinder  is  surmounted  by  a  graduated  circle,  M, 
upon  which  moves  a  pointer  or  index,  connected  at 
the  center  with  the  suspending  thread,  which  is 
twisted  when  the  index  is  turned.  The  lid,  C,  is 
perforated  with  a  hole  to  allow  access  to  the  pith 
ball  of  the  needle.  In  the  figure,  this  opening  is 
represented  as  closed  by  the  handle  of  a  movable  rod  of  glass  or 
lac,  which  insulates  the  ball  a,  by  which  electricity  is  conveyed 
to  the  ball  b  of  the  needle.  On  a  level  with  the  needle  is  a  cir* 
cular  band,  graduated  into  degrees  and  minutes.  It  is  usually 
made  of  paper,  and  pasted  around  the  cylinder. 

To  prepare  the  apparatus  for  experiments,  the  index  on  M  is 
set  opposite  to  zero,  and  then  the  circle,  conveying  the  index 
along  with  it,  is  turned,  until  the  ball  of  the  needle  rests  opposite 
to  zero,  on  its  graduated  circle.  In  this  situation,  the  suspending 
thread  is  entirely  untwisted,  or  free  from  torsion.  Now  let  the 
ball  a  be  electrified,  by  receiving  a  spark  from  the  prime  con- 
ductor, and  let  it  be  introduced  to  the  level  of  the  needle.  The 
ball  b  of  the  needle  being  unelectrified,  is  first  attracted  to  the 
electrified  ball,  imbibes  the  same  kind  of  electricity,  and  is  then 
repelled  to  a  greater  or  less  distance,  according  to  the  intensity 
of  the  electricity.  On  account  of  the  extreme  delicacy  of  the 
instrument,  only  a  very  small  charge  must  be  applied  ;  otherwise 
the  agitation  of  the  needle  will  be  in  danger  of  breaking  the 
thread,*  or  the  arc  described  by  the  needle  will  be  inconvenient- 
ly large.  The  charge  is  therefore  applied  from  a  pin's  head,  the 
pin  itself  being  concealed  in  sealing  wax.  The  pin's  head  being 
electrified,  it  is  touched  by  the  ball  a,  by  means  of  which,  the 
charge  is  introduced  into  the  cylinder  and  made  to  communicate 

*  The  filament  used  by  Coulomb  in  some  of  his  experiments,  was  a  silver  wire,  a 
foot  of  which  weighed  only  one  sixteenth  of  a  grain. 


ELECTRICITY.  397 

with  the  ball  b  of  the  needle.  Suppose  the  force  of  repulsion  be- 
tween the  two  balls  to  be  such,  that  the  needle  will  finally  settle 
at  the  distance  of  36°  from  zero,  or  the  point  where  it  was  qui- 
escent, it  would  describe  a  greater  arc  in  that  direction,  were 
not  its  motion  counteracted  by  the  force  of  torsion,  exerted  by 
the  suspending  wire. 

584.  Our  object,  it  will  be  recollected,  is  to  estimate  the  force 
of  this  repulsion  at  different  distances  from  the  electrified  ball  a. 
This  is  done  by  finding  the  relative  forces  of  torsion  required  to 
bring  those  respective  forces  of  repulsion  to  an  equilibrium.  We 
therefore  turn  the  index  upon  the  circle  M  in  a  direction  opposite 
to  that  in  which  the  needle  moved,  and  observe  the  number  of 
degrees  through  which  the  index  must  be  turned,  in  order  to 
make  the  ball  b  approach  to  any  given  distance  Fig.  217. 

of  the  ball  «.  Coulomb  proceeded  as  follows. 
The  ball  b  being  electrified  by.  contact  with  a, 
receded  from  it,  describing  an  arc  of  36°.  The 
index  on  the  circle  M  was  then  turned  in  the 
opposite  direction,  until  the  needle  was  carried 
back  to  the  distance  of  18°,  which  required 
the  index  to  be  turned  over  126°.  Again  the 
index  was  turned  until  the  needle  was  brought  to  the  distance 
of  8i°,  which  required  it  to  be  turned  over  567°.  Let  abd  re- 
present the  circle  in  which  these  movements  were  performed,  c 
being  its  center.  Take  ab  equal  to  36Q,  then  b  will  be  the  posi- 
tion of  the  needle  after  the  first  repulsion.  The  index  which 
carries  the  thread,  being  now  turned  backward  126°,  the  ball  b, 
were  it  free  to  move,  would  be  carried  over  the  same  arc  to  d', 
126°  beyond  a,  but  on  account  of  the  repulsion  of  the  ball  a,  it 
stops  short  at  b',  at  the  distance  of  18°  from  a.  Therefore,  the 
force  of  repulsion  of  the  two  balls  is  126°+180=1440.  In  the 
third  case,  where  the  index  was  turned  567°,  and  the  needle 
brought  to  the  distance  of  8|°  of  a,  were  it  not  for  the  repulsion 
between  the  balls,  the  needle  would  have  been  carried  567°  be- 
yond a  to  d,  but  stops  short  of  a  8i°  ;  therefore,  that  repulsion  is 
equal  to  567°+8£=575i°.  Hence,  the  respective  forces  of  re 
pulsion  exerted  at  the  several  distances,  were  as  follows  : 

36°         -        -        -        -       36  which  are  1  :     1. 

18°         -         ---     144        «         "    i  :     4. 

8ls°  -  -  -  -  575|  "  "  £  :  16,  nearly. 
It  appears  that  the  distances  are  to  one  another  nearly  in  the 
ratio  of  the  numbers  1,  |,  £,  while  the  corresponding  forces  are 
as  1,4,  16  ;  that  is,  the  force  of  repulsion  between  two  electrified 
bodies,  at  different  distances,  varies  inversely  dls  the  square  of  the 
distance.* 

*  Blot,  Prdcis  Elem.  tome  I,  482. 


398  NATURAL   PHILOSOPHY. 

The  same  law,  therefore,  governs  the  electrical  forces  as  that 
which  prevails  among  the  bodies  of  the  solar  system. 

585.  Analogous  experiments  prove  that  attraction  obeys  the 
same  law.  Some  practical  difficulty  was  experienced  by  Coulomb, 
in  his  experiments  on  attraction,  since,  when  the  balls  are  differ- 
ently electrified,  as  they  must  of  course  be  in  experiments  on  at- 
traction, they  will  come  together  if  brought  within  moderate 
distances  of  each  other.     But  the  law  was  satisfactorily  shown 
to  hold  good,  at  such  distances  as  were  susceptible  of  measure- 
ment, and  the  law  was  further  established  by  a  process  totally 
different  from  the  preceding.     It  consisted  in  bringing  the  sus- 
pended needle  near  to  an  insulated  electrified  sphere,  by  which 
it  is  made  to  oscillate  with  greater  or  less  rapidity,  according  to 
its  degree  of  proximity.     The  number  of  oscillations,  in  a  given 
time,  is  a  measure  of  the  force  of  attraction,  as  the  number  of 
oscillations  of  the  pendulum  measures  the  force  of  gravity,  being 
universally  as  the  square  root  of  the  forces.     (Art.  183.)     The 
proposition  may  therefore  be  stated  in  general  terms — 

The  force  of  electrical  attraction  or  repulsion,  at  different  distan- 
ces from  an  electrified  body,  varies  inversely  as  the  square  of  the 
distance. 

RATE    AT    WHICH    CHARGED    BODIES    LOSE    THEIR    ELECTRICITY. 

586.  It  is  a  well-known  fact,  that  when  an  insulated  conduct- 
or, charged  with  electricity,  is  suffered  to  remain  untouched  for 
a  certain  time,  it  will  gradually  lose  its  charge.     Now  since,  in 
some  of  the  delicate  researches  of  Coulomb,  a  considerable  time 
was  necessarily  occupied,  the  electrified  bodies  under  examina- 
tion might  change  their  degree  of  excitement  during  the  experi- 
ments, and  thus  give  a  fallacious  result.     It  became  important, 
therefore,  to  ascertain  the  law  according  to  which  this  dissipa- 
tion or  loss  of  electricity  took  place,  and  to  make  suitable  allow- 
ance for  it. 

Three  causes  chiefly  operate  in  depriving  a  body  under  these 
circumstances  of  its  electricity  : — first,  the  imperfection  of  bodies 
employed  as  insulators ;  secondly,  the  contact  of  successive  por- 
tions of  air,  every  particle  of  which  carries  off  a  certain  quantity 
of  the  fluid  ;  thirdly,  the  presence  of  moisture,  which  increases 
the  conducting  powers  of  all  surfaces.  (Art.  573.)  No  sub- 
stance is  actually  impervious  to  electricity  ;  that  is,  there  is  no 
substance  known,  of  which  any  portion,  however  small,  will  in- 
sulate perfectly  any  charge  however  great.  Still,  by  diminishing 
the  intensity  of  the  charge,  or  by  increasing  the  length  of  the 
substance  it  has  to  traverse,  a  degree  of  insulation  may  be  ob- 
tained in  which  the  escape  of  the  fluid  is  imperceptible.  This 
tendency  of  electricity  to  escape  from  charged  bodies,  is  inde- 


ELECTRICITY.  399 

pendent  of  the  chemical  nature  of  those  bodies,  being  the  same, 
under  similar  circumstances,  for  balls  of  wax,  copper,  elder  pith, 
and  various  other  substances.  The  same  tendency  is  equally 
independent  of  the  shape  and  magnitude  of  bodies,  unless  when 
the  intensity  of  the  charge  is  high  ;  in  which  case,  a  figure  that 
involves  points  and  edges  favors  the  dissipation  of  the  fluid. 
When  bodies  are  highly  charged,  the  electricity  is  lost  with  com- 
parative rapidity  ;  more  slowly  as  the  charge  is  less  ;  and  the 
air  being  dry,  and  the  insulator  of  a  proper  length,  a  certain 
charge  will  be  retained  without  further  loss.* 

But  the  chief  source  of  dissipation  of  the  electric  charge,  arises 
from  moisture,  either  existing  in  the  air,  or  settling  upon  the  sur- 
face of  the  insulating  supports,  or  imbibed  into  the  fibres  of  in- 
sulating threads. 

DISTRIBUTION    OP   ELECTRICITY. 

587.  Does  electricity  reside  only  at  the  surfaces  of  bodies,  or  is 
it  expanded  throughout  the  whole  of  their  substance  ?     Coating 
a  conductor  with  some  non-conducting  substance,  (as  a  wire  with 
sealing-wax,  leaving  the  ends  naked,)  does  not  in  the  least  im- 
pede the  passage  of  fluid  through  it.     Indeed,  every  conductor 
may  be  considered  as  really  in  this  situation,  being  in  contact 
with  a  stratum  of  air  on  every  side,  which,  when  dry,  is  a  good 
non-conductor.     The  conclusion  from  this  fact  is,  that  the  pass- 
age of  the  fluid  is  not  confined  to  the  surface,  mathematically 
considered,  but  must,  at  least,  occupy  the  exterior  stratum  of  the 
conductor.     It  was  found,  however,  by  Coulomb,  that  if,  of  two 
bodies  of  equal  surface  and  similar  form,  as  two  equal  spheres, 
one  be  electrified,  and  the  other  be  brought  into  contact  witn  it, 
the  electricity  will  be  equally  divided  between  them,  ancf  that 
this  takes  place  when  one  sphere  is  solid  and  the  other  hollow, 
equally  as  when  both  spheres  are  solid.     Hence  it  is  inferred, 
that  electricity  resides  only  at  or  very  near  the  surfaces  of  bodies,  f 

588.  This  fact  is  strikingly  illustrated  by  Fig.  218. 
an  experiment,  proposed  by  M.  Biot.  J    Let  S, 

(Pig.  218,)  represent  any  spheroid  of  con- 
ducting matter,  suspended  by  a  thread  of 
some  perfectly  insulating  substance.  Let  E, 
E,  be  two  caps  formed  of  gilt  paper,  tinfoil, 
or  any  other  conductor,  and  such  that  when  united,  they  accu- 

*  Lunn,  Encyc.  Metrop. 

t  Although  electricity  resides  only  at  the  surfaces  of  bodies,  yet  the  conducting 
power  of  a  body,  that  is,  its  power  to  transmit  a  charge,  is  proportioned  to  the  mass, 
or  quantity  of  matter.  (Faraday.) 

t  Precis  Ele"m.  tome  I,  p.  498. 


400  NATURAL   PHILOSOPHY. 

rately  fit  the  surface  of  the  spheroid.  An  insulating  handle  of- 
lac  is  also  attached  to  each  of  the  caps.  Now  let  there  be  com- 
municated to  the  ball  S,  any  degree  of  electricity,  and  then  «care- 
fully  apply  to  it  the  two  caps,  holding  them  by  their  insulating 
handles.  Upon  removing  these  caps,  it  will  be  found  that  every 
particle  of  electricity  has  been  abstracted  from  the  spheroid,  so 
that  it  will  no  longer  affect  the  most  delicate  electrometer ; 
while  the  two  caps  will  be  found,  upon  accurate  trial,  to  have 
acquired  precisely  the  same  quantity  of  electricity  that  before 
resided  upon  the  body  S. 

A  proof  of  this  point,  equally  conclusive,  and  applicable  to 
bodies  of  every  form,  was  devised  by  Coulomb.  An  insulated, 
solid  conductor,  of  any  figure,  being  provided,  cavities  were  dug 
in  it,  to  different  depths  below  the  surface,  and  in  several  different 
places,  and  the  body  was  electrified.  A  proof  plane,  as  it  was 
called,  consisting  of  a  small  circle  of  gilt  paper,  to  which  was  at- 
tached an  insulating  handle  of  lac,  was  introduced  into  these  va- 
rious cavities  at  different  depths.  It  was  then  withdrawn,  and 
tested  by  the  electrometer,  and  not  the  slightest  trace  of  electri- 
city was  indicated.  In  these  experiments,  care  was  taken  to  in- 
troduce the  proof  plane,  in  such  a  way  as  not  to  touch  the  edges 
of  the  cavities,  or  any  part  of  the  surface,  the  object  being  to 
ascertain  whether  signs  of  electricity  were  exhibited  at  any  depth 
below  the  surface.  The  conclusion  was,  that  there  were  none, 
and  consequently  that  the  electricity  of  excited  bodies  resides 
wholly  at  the  surface.* 

An  experiment,  which  may  be  easily  repeated,  shows  how 
much  the  intensity  of  an  electric  charge  is  affected  by  the  extent 
of  surface  which  it  pervades.  Let  a  sheet  of  tinfoil  be  wrapped 
several  times  around  an  insulated  cylinder,  which  is  mounted  so 
as  to  t^rn  horizontally  on  an  axis.  Upon  unwinding  the  metallic 
sheet,  and  thus  increasing  the  extent  of  electrified  surface,  an 
electrometer  connected  with  the  cylinder  will  indicate  a  decline 
in  the  intensity  of  the  charge,  at  every  successive  enlargement 
of  surface. 

589.  Although  electricity  resides  at  the  surface  of  an  electrified 
body,  yet  it  is  not  distributed  uniformly  over  that  surface,  except 
the  body  be  a  perfect  sphere,  but  is  unequally  accumulated,  in 
different  parts  of  the  surface,  m  a  manner  depending  on  the  fig- 
ure of  the  body.  The  principle  may  be  enunciated  in  general 
terms,  thus : — 

In  conductors  of  an  elongated  fgure,  the  electricity  is  accumu- 
lated toward  the  two  ends,  and  withdrawn  more  or  less  from  the 
central  parts. 

Coulomb,  in  his  investigations  on  this  subject,  employed  the 

*  Biot,  Precis  El.  I,  p.  500 


ELECTRICITY.  401 

proof  plane,  (Art.  588,)  the  circle  of  gilt  paper  being  so  small 
as  to  bear  no  considerable  ratio  to  the  surface  of  the  electrified 
body  under  examination.  By  touching  this  plane  to  different 
points  of  the  surface,  the  plane  imbibes  the  charge  belonging  to 
that  point,  and  may  be  made  to  transfer  it  to  the  balls  of  the  elec- 
trical balance.  (Fig.  216.)  Then  the  amount  of  torsion  required 
to  bring  the  balls  to  the  same  given  distance  of  each  other,  will 
be  a  measure  of  the  charge  communicated  to  the  balls  in  each 
case  ;  that  is,  the  torsions  will  indicate  the  ratios  existing  between 
the  different  charges  of  electricity,  at  different  points  in  the  sur- 
face of  the  body  under  examination. 

In  this  manner,  Coulomb  determined  the  distribution  of  elec- 
tricity upon  a  steel  plate,  eleven  inches  long,  one  inch  broad, 
and  half  a  line  thick,  insulated  and  electrified.  In  order  to  cover 
the  breadth  of  the  plate,  the  gilt  paper  was  made  an  inch  long, 
but  very  narrow.  First,  the  proof  plane  was  applied  to  the  cen- 
ter of  the  plate,  and  at  one  inch  from  the  extremity  ;  the  latter 
charge  was  to  the  former  as  1.2  to  1,  and  therefore  nearly  equal. 
Secondly,  on  applying  the  plane  quite  at  the  extremity,  the 
charge  was  to  that  at  the  center  as  2  to  1.  Thirdly,  the  plane 
was  applied,  at  one  end,  to  the  extreme  edge,  so  as  to  be  in  con- 
tact with  both  surfaces ;  in  which  case,  the  charge  was  double 
that  of  each  extreme  surface,  and,  of  course,  four  times  that  of 
the  central  parts. 

590.  Hence  it  appears,  that  the  electricity  of  a  conductor, 
analogous  to  the  steel  plate  employed  in  the  foregoing  experiments, 
is  nearly  uniform  on  all  parts  of  the  surface,  except  the  two  ends, 
where  it  becomes  twice  as  great  as  in  the  other  parts.  The  rapid 
increase  of  electricity  toward  the  extremities,  appears  also  in 
other  bodies  of  an  elongated  figure  ;  and  the  augmentation  is  the 
more  rapid,  as  the  length  is  greater  in  respect  to  the  diameter ; 
and  when  the  extremity  becomes  elongated,  like  the  point  of  a 
cone,  the  accumulation  at  that  extremity  becomes  so  great,  that 
the  resistance  of  the  air  is  not  sufficient  to  retain  it,  and  it  escapes, 
producing  the  electric  spark.  Hence  the  reason  why  points, 
connected  with  an  electrified  conductor,  dissipate  the  fluid  so 
rapidly. 

The  limited  extent  of  this  work,  does  not  permit  us  to  give  a 
more  particular  account  of  the  researches  of  Coulomb,  carried  on 
by  the  aid  of  the  torsion  balance  ;  but  we  would  recommend  these 
researches,  as  detailed  by  Biot,*  to  the  student  of  natural  philos- 
ophy, as  examples  of  the  most  refined,  ingenious,  and  conclusive 
experiments. 

*  Precis  Elementaire  de  Physique,  tome  I. 
51 


402  NATURAL   PHILOSOPHY. 

CHAPTER  HI. 

OF  THE  LEYDEN  JAR. 

591.  THIS  instrument,  which  is  a  very  important  and  interest- 
ing article  of  electrical  apparatus,  consists  of  a  glass  jar,  coated 
on  both  sides  with  tinfoil,  except  a  space  on  the  upper  end,  with- 
in two  or  three  inches  of  the  top,  which  is  either  left  bare,  or  is 
covered  with  a  coating  of  varnish,  or  a  thin  layer  of  sealing 
wax.  To  the  mouth  of  the  jar  is  fitted  a  cover  of  hard  baked 
wood,  through  the  center  of  which  passes  a  perpendicular  wire, 
Fig.  219.  Fig.  220. 


terminating  above  in  a  knob,  and  below  in  a  fine  chain,  that 
rests  upon  the  bottom  of  the  jar.  On  presenting  the  knob  of  the 
jar  near  to  the  prime  conductor  of  an  electrical  machine,  while 
the  latter  is  in  operation,  a  series  of  sparks  passes  between  the 
conductor  and  the  jar,  which  will  gradually  grow  more  and  more 
feeble,  until  they  cease  altogether.  The  jar  is  then  said  to  be 
charged.  If  now  we  take  the  discharging  rod,  (which  is  a 
crooked  wire,  armed  at  each  end  with  knobs,  and  insulated  by  a 
glass  handle,  as  in  Fig.  220,)  and  apply  one  of  the  knobs  to  the 
outer  coating  of  the  jar,  and  bring  the  other  to  the  knob  of  the 
jar,  a  flash  of  intense  brightness,  accompanied  by  a  loud  report, 
immediately  ensues.  On  applying  the  discharging  rod  a  second 
time,  a  feeble  spark  passes,  being  the  residuary  charge,  after 
which  all  signs  of  electricity  disappear,  and  the  jar  is  said  to  be 
discharged. 

If,  instead  of  the  discharging  rod,  we  apply  one  hand  to  the 
outside  of  the  charged  jar,  and  bring  a  knuckle  of  the  other 
hand  to  the  knob  of  the  jar,  a  sudden  and  surprising  shock  is 
felt,  convulsing  the  arms,  and,  when  sufficiently  powerful,  pass- 
ing through  the  breast. 

592.  The  Leyden  jar  derives  its  name  from  the  place  of  its 
discovery.  In  the  year  1746,  while  some  philosophers  of  Leyden 


ELECTRICITY.  403 

were  performing  electrical  experiments,  one  of  them  happened 
to  hold,  in  his  hand,  a  tumbler  partly  filled  with  water,  to  a  wire 
connected  with  the  prime  conductor  of  an  electrical  machine. 
When  the  water  was  supposed  to  be  sufficiently  electrified,  he 
attempted,  with  the  other  hand,  to  detach  the  wire  from  the  ma- 
chine ;  but  as  soon  as  he  touched  it,  he  received  the  electric 
shock.  It  was  by  imitating  this  arrangement,  that  the  Leyden 
jar  was  constructed  ;  for  here  was  a  glass  cylinder,  having  good 
conductors  on  both  sides,  viz.  the  hand  on  the  outside,  and  water 
on  the  inside,  which  were  prevented  from  communicating  with 
each  other  by  the  non-conducting  powers  of  the  glass.  A  me- 
tallic coating,  as  tinfoil  or  sheet  lead,  was  substituted  for  the 
two  conductors,  and  a  jar  for  the  tumbler,  and  thus  the  electrical 
jar  was  constructed. 

593.  Those  who  first  received  the  electric  shock  from  the 
Leyden  jar,  gave  the  most  extravagant  account  of  its  effects. 
M.  Muschenbroeck,  a  philosopher  of  Leyden,  of  much  eminence, 
said  that  "  he  felt  himself  struck  in  his  arms,  shoulders,  and 
breast,  so  that  he  lost  his  breath,  and  it  was  two  days  before  he 
recovered  from  the  effects  of  the  blow  and  the  terror ;  adding, 
that  he  would  not  take  a  second  shock  for  the  kingdom  of 
France."  M.  Winkler,  of  Leipsic,  testified,  that  "  the  first  time 
he  tried  the  Leyden  experiment  he  found  great  convulsions  by  it 
in  his  body  ;  and  that  it  put  his  blood  into  great  agitation,  so 
that  he  was  afraid  of  an  ardent  fever,  and  was  obliged  to  use 
refrigerating  medicines.  He  also  felt  a  heaviness  in  his  head, 
as  if  a  stone  lay  upon  it,  and  twice  it  gave  him  a  bleeding  at  the 
nose." 

In  an  age  less  enlightened  than  the  present,  and  less  famil- 
iar with  the  wonders  of  philosophy  and  chemistry,  the  striking 
and  truly  surprising  effects  of  electricity,  as  exhibited  by  the 
Leyden  jar,  would  naturally  excite  great  admiration  and  aston- 
ishment. Accordingly,  showmen  travelled  with  this  apparatus 
through  the  principal  cities  of  Europe,  and  probably  no  object 
of  philosophical  curiosity  ever  drew  together  greater  crowds  of 
spectators.  It  was  this  astonishing  experiment,  (says  Dr.  Priest- 
ley,) that  gave  eclat  to  electricity.  From  this  time  it  became 
the  subject  of  general  conversation.  Everybody  was  eager  to 
see,  and,  notwithstanding  the  terrible  account  that  was  reported 
of  it,  to  feel  the  experiment ;  and  in  the  same  year  in  which  it 
was  discovered,  numbers  of  persons,  in  almost  every  country  in 
Europe,  got  a  livelihood  by  going  about  and  showing  it.  All 
the  electricians  of  Europe, «lso,  were  immediately  employed  in 
repeating  this  great  experiment,  and  in  attending  to  the  circum- 
stances of  it.*  With  similar  assiduity,  and  unequalled  success, 

*  Priestley's  Hist.  Elec.,  p.  84. 


404  NATURAL   PHILOSOPHY. 

Dr.  Franklin  betook  himself  to  experiments  on  the  Leyden  jar. 
He  effectually  investigated  all  its  properties,  by  very  diversified 
and  ingenious  experiments,  and  gave  the  first  rational  explana- 
tion of  the  cause  of  its  phenomena.  The  following  experiments 
may  be  easily  repeated. 

594.  (1.)  The  jar  is  charged  by  bringing  the  knob  near  to  the 
prime  conductor,  while  the  machine  is  in  operation.  One  mode  of 
charging  the  jar  has  been  already  mentioned  in  Art.  591.  It 
may,  however,  either  be  held  in  the  hand,  or  placed  on  the  table, 
or  on  any  conducting  support :  the  only  circumstance  to  be  at- 
tended to  is,  that  the  outside  shall  be  uninsulated.  A  jar,  while 
charging,  will  sometimes  discharge  itself  spontaneously.  This 
effect  will  be  more  likely  to  happen,  if  the  uncoated  interval  is 
very  clean  and  dry,  and  may  be  prevented  altogether,  by  previ- 
ously breathing  on  the  uncoated  part.* 

(2.)  The  opposite  sides  of  a  charged  jar  are  in  different  electric- 
al states,  the  one  positive  and  the  other  negative.  Thus,  if  a  pith 
ball,  suspended  by  a  silk  thread,  be  applied  to  the  knob,  it  will 
first  be  attracted  to  it,  and  then  repelled  ;  but  it  will  now  be  at- 
tracted by  the  outside  coating,  until  it  becomes  electrified  in  the 
same  way,  and  then  repelled,  and  so  on. 

(3.)  In  order  to  receive  the  charge,  the  outside  of  the  jar  must  be 
uninsulated.  If  we  attach  a  string  to  the  knob  of  the  jar,  and 
suspend  the  jar  in  the  air,  to  the  prime  conductor,  and  put  the 
machine  in  operation,  no  charge  will  be  communicated  to  the 
jar.  The  same  result  will  follow,  if  the  jar  stands  on  an  insu- 
lated stand,f  or  is  insulated  by  any  other  method.  An  insulated 
jar,  however,  may  be  charged  by  connecting  its  knob  with  the 
positive  conductor,  and  its  outer  coating  with  the  rubber. J 

(4.)  A  second  jar  may  be  charged,  by  communication  with  the 
outside  of  the  jirst,  while  the  latter  is  receiving  its  charge.  The 
charge  communicated  to  the  second  jar,  is  of  the  same  kind  as 
that  of  the  first,  and  nearly  of  the  same  degree  of  intensity,  pro- 
vided the  capacity  of  the  two  jars  be  the  same.  Moreover,  if  a 
third,  a  fourth,  or  any  number  of  jars  of  the  same  size,  be  con- 
nected in  a  similar  manner  with  each  other,  namely,  having  the 
knob  of  each  in  communication  with  the  outside  coating  of  the 
next  preceding, — then  all  the  jars  will  be  charged  with  the  same 
kind  of  electricity,  but  the  degree  of  intensity  will  decline  a  little 
in  v  the  successive  jars.  If  the  charge  be  derived,  through  the 

*  Singer,  El.  Elec.,  p.  101. 

t  An  insulating  stand  is  any  flat  support,  instated  by  a  pillar  of  glass.  The  pillar 
is  usually  a  solid  cylinder  of  glass,  from  six  to  twelve  inches  long,  varnished  so  as  to 
protect  it  from  moisture.  A  junk  bottle,  surmounted  by  a  circular  piece  of  polished 
wood,  dry  and  varnished,  makes  a  very  good  insulating  support 

t  Singer,  El.  Elec.,  p.  106. 


ELECTRICITY.  405 

prime  conductor,  from  the  cylinder  or  plate,  as  is  usually  the 
case,  it  will  be  the  positive,  or  vitreous  electricity. 

(5.)  A  jar  may  be  charged  negatively,  by  receiving  the  electricity 
of  the  rubber, — the  rubber  being  insulated,  and  the  prime  con- 
ductor uninsulated.  For  this  purpose,  the  chain  usually  attached 
to  the  rubber  may  be  transferred  to  the  prime  conductor.  Also, 
a  jar  may  be  charged  negatively,  by  grasping  the  jar  by  the 
knob,  and  receiving  the  electricity  of  the  prime  conductor  on  the 
outside.  It  must  be  set  down  on  an  insulated  support,  else  the 
operator  will  receive  a  shock. 

(6.)  When  two  jars  are  charged,  the  one  positively  and  the  other 
negatively,  on  forming  a  communication  between  the  insides  of  both, 
by  connecting  the  two  knobs,  no  discharge  will  take  place,  unless  the 
outsides  be  in  conducting  communication.  Thus,  if  two  jars  be 
charged,  the  one  from  the  prime  conductor  and  the  other  from 
the  rubber,*  and  placed  at  the  distance  of  a  few  inches  from 
each  other,  on  insulated  supports,  on  connecting  the  two  knobs 
by  the  discharging  rod,  no  discharge  will  follow ;  but,  let  a  wire 
be  laid  across  the  supports,  touching  the  outside  of  each  jar ; 
then,  on  applying  the  discharging  rod  to  the  two  knobs,  an  ex- 
plosion will  immediately  ensue. 

By  means  of  two  jars  differently  charged,  and  placed  as  above, 
with  their  outsides  in  conducting  communication,  the  experiment 
may  be  exhibited,  which  is  called  the  Electrical  Spider.  It  con- 
sists of  a  small  piece  of  cork,  so  fashioned  as  to  represent  the 
body  of  a  spider,  and  blackened  with  ink,  having  a  number  of 
black  linen  threads  drawn  through  it  to  represent  the  legs.  This 
is  suspended  by  a  silk  thread,  half  way  between  the  knobs  of  the 
two  jars,  and  vibrates  for  a  long  time  from  one  knob  to  the  other, 
until  both  jars  are  discharged.  The  rationale  will  be  obvious  on 
a  little  reflection. 

(7.)  The  charge  of  any  jar  may  be  divided  into  definite  parts  ; 
that  is,  the  half,  the  fourth,  or  any  aliquot  part  of  the  charge 
may  be  taken,  f  This  may  be  done  by  connecting  the  inner  and 
outer  coating  of  the  charged  jar,  with  the  inner  and  outer  coating 
of  an  unelectrified  jar,  of  the  same  size  and  thickness.  The 
respective  charges  will  be  measured  by  the  quadrant  elec 
trometer.J  (Fig.  214.)  * 

(8.)  The  electricity  is  accumulated  on  the  surface  of  the  glass, 
and  the  coatings  serve  merely  as  conductors  of  the  charge.  This 
is  proved  by  the  fact  that  when  the  coatings  are  movable,  so  that 

*  And  both  may  be  thus  charged  at  the  same  time,  by  connecting  one  with  the 
insulated  rubber,  and  the  other  with  the  insulated  prune  conductor,  the  jars  them- 
selves being  uninsulated. 

t  Singer,  p.  110. 

t  It  is  essential,  however,  that  the  electrometer  should  be  graduated,  not  by  equal 
divisions,  but  according  tf>  a  scale  of  arcs,  the  tangents  of  which  are  in  arithmetical 
progression 


406  NATURAL   PHILOSOPHY. 

they  can  be  taken  off  from  the  jar  after  it  is  charged,  neither  of 
them  exhibits  the  least  sign  of  electricity ;  while  if  another  pair 
of  coatings  is  substituted,  which  have  not  been  electrified,  on 
forming  the  communication-  between  the  inside  and  outside,  the 
usual  discharge  takes  place,  showing  that  the  whole  of  the  charge 
was  retained  on  the  glass  surfaces  of  the  jar.* 

(9.)  The  charge  of  a  Leyden  jar  may  be  retained  for  a  long 
time.  If  the  surfaces  .are  well  separated  from  each*  other,  the 
charge  remains  for  many  days,  or  even  weeks.  The  charge  is 
usually  dissipated  by  the  motion  of  particles  of  dust,  or  other 
conducting  substances  in  the  atmosphere,  from  one  of  the  coatings 
to  the  other,  or  by  the  uncoated  interval  becoming  moist  and 
losing  its  insulating  power  ;  consequently  a  jar  will  retain  its 
charge  longer  in  dry  than  in  damp  weather.  Covering  the 
uncoated  part  of  the  jar  with  melted  sealing  wax  or  varnish, 
prevents  the  deposition  of  moisture  upon  it,  and  consequently 
tends  also  materially  to  prevent  the  dissipation  of  its  charge.f 

(10.)  A  pane  of  glass,  a  plate  of  air,  or  any  other  similar  elec- 
tric, may  be  charged  to  a  greater  or  less  degree  in  a  manner  anal- 
ogous to  that  of  the  Leyden  jar. — If  a  pane  of  glass  is  coated  on 
both  sides  with  a  sheet  of  tinfoil,  leaving  an  uncoated  interval 
all  round  the  edges  for  the  space  of  two  inches ; — and  if  we  then 
hold  the  pane  by  one  corner  and  apply  the  knuckle  to  the  outer 
coating,  and  bring  the  inner  coating  to  the  prime  conductor,  the 
pane  will  be  charged,  and  may  be  discharged,  by  applying  the 
knobs  of  the  discharging  rod  to  the  opposite  metallic  coatings. 
A  plate  of  air  may  be  charged  in  the  same  manner  as  a  plate  of 
glass ;  but  as  air  is  more  readily  displaced  by  electricity,  in 
consequence  of  the  mobility  of  its  particles,  a  thicker  stratum  of 
it  must  be  employed.  The  usual  form  of  the  experiment  is  to 
employ  two  circular  disks  of  wood  covered  with  tinfoil,  and  well 
rounded  at  the  edges,  having  a  diameter  of  from  two  to  four  feet. 
One  of  the  boards  is  to  be  placed  flat  upon  a  table,  and  the  other 
being  suspended  by  a  silk  cord  from  the  ceiling,  is  adjusted  so 
as  to  hang  parallel  over  its  surface,  and  at  the  distance  of  an 
inch  or  an  inch  and  a  half  from  it.  The  upper  insulated  board 
being  connected  with  an  electrical  machine,  the  stratum  of  air 
between  th'j  boards  becomes  charged,  and  will  communicate  a 
shock  if  the  upper  and  lower  one  be  touched  at  the  same  time 
with  opposite  hands.  The  shock  produced  in  this  way  is  con- 
siderably less  violent  than  that  from  an  equal  surface  of  coated 
glass;  for  the  distance  of  the  coatings  is  of  necessity  much 
greater,  and  the  medium  between  them  less  perfectly  insulating ; 
and  this  last  circumstance  operates  so  rapidly  when  the  charge 
is  high,  that  its  maximum  of  effect  cannot  be  obtained  but  by 
making  the  discharge  while  the  machine  is  in  action.  If  the 

*  Singer,  p.  112.  t  Ib.  116. 


ELECTRICITY.  407 

discharge  is  not  made,  spontaneous  explosions  from  one  disk  to 
the  other,  through  the  intervening  plate  of  air,  will  occur  at 
intervals,  as  long  as  the  electrization  of  the  upper  disk  is  con- 
tinued. 

(11.)  If  a  coated  pane  of  glass  be  held  vertically,  with  two  of 
its  edges  parallel  with  the  horizon,  and  to  the  upper  edges  of  the 
metallic  coating  two  threads  be  attached  directly  opposite  to 
each  other;  on  communicating  a  spark  to  one  of  the  coatings, 
the  two  threads  both  rise,  forming  equal  angles  with  the  surface 
of  the  glass.  On  applying  a  conductor,  as  the  finger,  to  one  of 
the  coatings,  the  thread  on  that  side  immediately  falls,  while  the 
other  thread  doubles  its  angle  of  elevation ;  so  that  the  angles 
intercepted  between  the  two  threads,  is  a  constant  quantity.* 

Before  the  learner  is  qualified  to  understand  the  explanation 
of  the  foregoing  experiments,  he  must  become  more  fully  ac- 
quainted with  the  law  of  induction,  (Art.  575,)  upon  which  the 
theory  of  the  Leyden  jar  depends. 

LAW   OF   INDUCTION,  f 

595.  Active  electricity,  existing  in  any  substance,  tends  always 
to  induce  the  opposite  electrical  state  in  the  bodies  that  are  near 
to  it.  It  is  our  object,  in  this  section,  to  exhibit  this  important 
principle  more  fully  than  has  yet  been  done  in  the  preceding 
pages. 

Let  A  (Fig.  221,)  represent  an  elec-  Fig.  221. 

trical  glass  globe,  and  B  a  metallic  cyl-  Q^l3LJ  )  -Jp  (A) 
inder,  placed  on  insulating  supports,  Aft  <&  A  A  ]|</V 
near  to  the  glass  globe,  but  not  near 
enough  for  a  spark  to  pass.  To  the  cyl- 
inder, let  five  pairs  of  pith  balls  be  sus- 
pended, by  conducting  threads,  viz.  one  pair  near  each  end,  one 
near  the  center,  and  one  about  half  way  between  the  center  and 
either  extremity.  We  shall  find  that  every  pair  of  pith  balls, 
except  those  situated  at  a  particular  part  of  the  cylinder  not  far 
from  the  center,  will  immediately  diverge,  indicating  the  elec- 
trical state  of  the  part  from  which  they  are  suspended.  Those 
at  either  extremity  diverge  most ;  and  the  divergence  diminishes 
as  we  approach  the  central  parts  to  a  certain  point,  where  the 
pith  balls  suffer  no  effect,  and  where,  consequently,  the  body  is 
in  its  natural  state.  By  means  of  the  electrometer,  we  may 
ascertain  that  the  species  of  electricity  is  negative,  or  opposite 
to  that  of  the  glass  globe,  in  all  those  parts  of  the  cylinder  which 
are  nearer  to  the  globe  than  the  before  mentioned  neutral  point ; 
and  that  it  is  positive  in  all  parts  of  the  cylinder  more  distant 

*  ED  eye.  Metropolf)  Elec.,  p.  92. 

t  Biot,  Precis  Elem.  tome  I.,  or  Library  of  Useful  Knowledge,  Art.  Electricity. 


408  NATURAL   PHILOSOPHY. 

than  this  point.  We  may  ascertain  with  much  greater  accu- 
racy these  electrical  states,  by  the  employment  of  the  proof  plane 
and  electrometer  of  Coulomb,  (Art.  589,)  than  by  pith  balls ; 
and  the  results  are  then  found  to  correspond  with  the  results  of 
theory,  to  be  stated  hereafter. 

596.  These  effects,  it  should  be  remarked,  are  simply  the  re- 
sult of  electrical  action  •  at  a  distance  ;  for  they  depend  upon  no 
other  circumstance.     They  take  place  in  an  equal  degree,  what- 
ever substance  is  interposed  between  the  bodies  which  are  exert- 
ing this  action  on  one  another,  provided  the  interposed  substance 
undergoes  no  change  in  its  own  electrical  state  ;  a  condition 
which  is  fulfilled  in  electrics,  or  non-conducting  bodies  only. 
Thus,  induction  will  take  place  just  as  effectually  through  a 
plate  of  glass,  as  if  no  such  substance  had  intervened. 

Let  us  now  suppose  that  the  acting  body  A  is  not  glass,  or  any 
electric,  but  a  conducting  body,  a  sphere  of  copper,  for  example, 
charged  with  positive  electricity,  and  insulated  on  a  glass  support. 
The  primary  effects  of  this  sphere  on  the  cylinder  will  be  the 
same  as  in  the  former  case ;  but  the  electrical  state  which  the 
cylinder  has  acquired  at  the  end  adjacent  to  the  globe,  will  react 
upon  the  electricity  of  the  globe,  tending  to  put  it  into  a  state  still 
further  opposite  to  its  own,  that  is,  to  render  the  nearer  parts  of 
the  globe  positive  in  a  higher  degree  than  they  were  before. 
This  can  be  done  only  at  the  expense  of  the  other  side  of  the 
globe,  which  thus  becomes  less  positive  than  before.  But  this 
new  distribution  of  the  electric  fluid  in  the  globe,  by  increasing 
the  positive  state  of  the  side  next  to  the  cylinder,  tends  to  aug- 
ment its  inductive  influence  upon  the  fluid  in  the  cylinder  ;  that 
is,  to  drive  out  an  additional  quantity  of  the  fluid  from  the  nega- 
tive to  the  positive  end.  This  is  followed  in  its  turn  by  a  corres- 
ponding reaction  on  the  globe,  and  so  on,  constituting  a  series  of 
smaller  adjustments,  until  a  perfect  equilibrium  is  established  in 
every  part.  When  this  has  been  attained,  the  electrical  states 
will,  it  is  evident,  be  of  the  same  kind  as  those  consequent  upon 
the  immediate  actions,  though  somewhat  increased  in  intensity  by 
the  series  of  reactions. 

597.  The  following  experiment  is  a  practical  illustration  of  the 
preceding  remarks.     Furnish  the  copper  globe  with  a  pair  of  pith 
balls  on  each  of  two  opposite  sides.     When  the  globe  is  insulated 
and  alone,  any  electricity  communicated  to  it  will  diffuse  itself 
equally  over  the  surface,  and  both  pairs  of  balls  will  diverge 
equally.     But  on  bringing  near  to  it  a  conducting  body,  the  balls 
on  the  remoter  side  will  immediately  begin  to  collapse,  while 
those  at  the  nearer  side  diverge  to  a  greater  degree  than  before  ; 
thus  showing  the  nature  of  the  reflex  operation  of  the  induced 


ELECTRICITY.  409 

electricity  of  the  conductor,  upon  the  body  from  which  the  in- 
duction originated. 

It  should  be  recollected,  that  in  all  the  changes  we  have  thus 
traced  as  the  effects  of  induction,  there  has  been  no  transfer  of 
electricity  from  either  of  the  bodies  to  the  other  ;  as  might  be  in- 
ferred from  their  taking  place  equally  well  when  a  plate  of  glass 
is  interposed.  Another  proof  is  afforded  by  the  circumstance,  that 
the  mere  removal  of  the  bodies  to  a  distance  from  one  another,  is 
sufficient  to  restore  each  of  them  to  its  original  state.  The  globe 
remains  as  positively  electrified  as  before  ;  the  cylinder  returns  to 
its  condition  of  perfect  neutrality ;  nothing  has  been  lost  and 
nothing  gained  on  either  side.  The  experiment  may  be  repeated 
as  often  as  we  please,  without  any  variation  of  the  phenomena. 
But  this  would  not  be  the  case  if  the  cylinder  were  divided  in  the 
middle,  and  one  or  both  of  the  parts  were  removed  separately, 
while  they  still  remained  under  the  influence  of  the  globe.  The 
return  of  the  electric  fluid  from  the  positive  to  the  negative  end 
being  thus  prevented,  each  part  will  retain,  after  its  separation, 
the  electricity  which  had  been  induced  upon  it.  The  nearer  por- 
tion will  remain  negative  ;  the  remoter  portion  positive.  If  the 
division  had  been  in  three  parts,  the  middle  part  only  would  have 
been  neutral.  The  experiment  may  be  made  by  joining  two  or 
more  conductors  endwise,  similar  to  B,  (Fig.  221,)  so  that  they 
may  act  as  a  single  conductor  when  placed  near  to  the  electrified 
globe,  and,  after  induction  has  thus  been  produced,  removing  them 
separately,  and  examining  their  electrical  states.  If  the  number 
of  conductors  be  three,  the  first  will  be  found  negative,  the  third 
positive,  and  the  second  neutral. 

598.  Another  modification  of  effect  will  take  place  when  an 
insulated  conductor,  rendered  electrical  at  both  ends  by  induction, 
is  made  to  communicate  with  another  insulated  conductor.  Let 
us  first  suppose  that  a  long  metallic  conductor  is  brought  into  con- 
tact with  the  remote  end  of  the  first  cylinder  B,  (Fig.  221,)  which 
has  been  rendered  positive  by  induction.  The  fluid  accumulated 
at  this  end  will  now  pass  into  the  conductor,  and  will  remove  to  the 
most  distant  part  of  it.  The  transit  will  take  place  before  actual 
contact,  and  will  be  manifested  by  the  appearance  of  a  spark, 
when  the  bodies  are  brought  within  the  striking  distance.  The  re- 
moval of  this  portion  of  fluid  to  a  greater  distance,  will  occasion 
a  disturbance  in  the  equilibrium  that  had  before  been  established. 
The  repulsion  which  that  fluid  had  excited,  and  which  had  con- 
tributed to  prevent  any  more  fluid  from  being  repelled  from  the 
negative  end,  is  now  considerably  weakened  by  the  greater  dis- 
tance at  which  it  acts  ;  and  more  fluid  will  leave  the  negative 
end,  which  end  will  consequently  become  more  highly  negative. 
This  change  of  distribution  will  again  occasion  a  further  effect, 
by  its  reaction  on  the  fluid  in  the  globe  whence  the  action  origi- 

52 


410  NATURAL   PHILOSOPHY, 

nally  proceeded  ;  and  another  series  of  changes  and  adjustments 
will  follow,  until  a  new  condition  of  equilibrium  takes  place,  and 
then  the  fluid  will  be  at  rest. 

599.  Thus  we  learn  that  the  effects  of  induction  in  a  conduct- 
or are  augmented  by  increasing  its  length  ;  they  would  therefore 
be  greatest  of  all,  if  we  could  give  it  infinite  length ;  but  the 
same  condition  is  attainable  by  placing  the  conductor  in  commu 
nication  with  the  earth,  which  will  carry  off  all  the  fluid  which  the 
electrified  body  is  capable  of  expelling  from  the  nearest  end.    Ac- 
cordingly, if  we  touch  with  the  finger,  or  with  a  metallic  rod  held 
in  the  hand,  the  remote  end  of  an  insulated  conductor  under  the 
influence  of  induction,  we  obtain  a  spark  more  or  less  vivid  ac- 
cording to  the  intensity  of  the  electricity  so  induced ;  and  the 
conductor  so  touched  has  now  only  one  kind  of  electricity,  name- 
ly, the  one  opposite  to  that  of  the  electrified  body  which  is  acting 
upon  it.    The  part  touched  is  brought  into  a  state  in  which  it  ap- 
pears to  be  neutral,  as  long  as  it  remains  in  the  vicinity  of  the 
electrified  body ;  because  the  actions  of  the  redundant  fluid  and 
unsaturated  matter  in  the  two  bodies,  exactly  balance  one  anoth- 
er.    But  it  all  the  while  really  contains  less  fluid  than  its  natural 
share,  in  consequence  of  the  repulsive  tendency  of  the  fluid  in 
the  body  which  produces  the  induction  ;  and  this  negative  state 
will  readily  become  active  if  the  conductor  that  has  been  touched 
be  again  insulated,  and  then  removed  from  the  influence  of  the 
former.     This  peculiar  condition  of  a  body,  in  which  its  parts 
are  really  undercharged  or  overcharged  with  fluid,  although,  from 
the  action  of  electrical  forces  derived  from  bodies  in  its  vicinity, 
a  state  of  equilibrium  is  established,  and  no  visible  effect  results, 
has  been  denominated  by  Biot  disguised  electricity. 

600.  We  have  hitherto  supposed  the  acting  body  to  be  posi- 
tively electrified ;  but  precisely  the  same  effects  would  happen 
with  regard  to  degree,  although  opposite  as  to  the  species  of  elec- 
tricity, if  it  had  been  negatively  electrified :  and  the  same  expla- 
nations will  in  every  respect  apply,  with  the  requisite  substitution 
of  the  terms  negative  for  positive,  and  of  attraction  for  repulsion, 
and  vice  versa.     A  little  reflection  will  also  show  the  application 
of  the  theory  of  double  electricities  to  explain  the  same  phenom- 
ena.    Calling  the  electricity  of  the  globe  vitreous  instead  of  posi- 
tive, and  substituting  the  term  resinous  for  negative,  we  then  say 
that  the  vitreous  electricity  of  the  globe  drives  oif  the  similar 
electricity  from  the  contiguous  end  of  the  .cylinder,  and  attracts 
to  it  the  resinous  fluid.     This  again  attracts  the  vitreous  fluid 
from  the  remoter  parts  of  the  globe  to  the  nearest  surface  ;  and 
thus,  the  vitreous  and  resinous,  instead  of  the  positive  and  nega- 
tive fluids,  act  and  react  on  each  other. 


ELECTRICITY.  411 

601.  Another  consequence  of  the  induction  of  electricity  must 
not  be  overlooked,  namely,  that  the  bodies  between  which  it  takes 
place,  necessarily  attract  one  another :  for  the  mutual  actions  be- 
tween the  contiguous  surfaces  of  the  globe  and  the  cylinder, 
(Fig.  221,)  which  are  in  opposite  electrical  states,  exceed  that  of 
the  remoter  surfaces  of  those  two  bodies  which  are  in  the  same 
electrical  state,  because  the  latter  surfaces  are  more  distant  from 
each  other  than  the  former,  and  the  force  of  electrical  action  is 
inversely  as  the  square  of  the  distance.     Hence  the  attractive 
force  always  exceeds  the  repulsive.    We  have  already  seen  (Art. 
575,)  that  this  circumstance  sufficiently  explains  the  fact,  that 
conducting  bodies  previously  neutral,  are  attracted  by  electrified 
bodies.     Another  fact,  which  appears  more  singular,  and  which 
cannot  be  accounted  for  on  any  other  principle,  is  also  a  direct 
consequence  of  the  law  of  induction.     If  a  small  insulated  body, 
weakly  electrified,  be  placed  at  a  distance  from  another  and  larger 
body  more  highly  charged  with  the  same  species  of  electricity, 
it  will,  as  usual,  be  repelled  ;  but  there  is  a  certain  distance,  with- 
in which  if  it  be  brought,  attraction  will  take  place  instead  of  re- 
pulsion.    This  happens  in  consequence  of  the  inductive  influ- 
ence producing  so  great  a  change  in  the  distribution  of  electricity, 
as  to  give  a  preponderance  to  the  attractive  forces  of  the  adja- 
cent parts  of  the  two  bodies,  over  the  repulsive  forces  that  take 
place  in  the  other  parts,  and  which  would  have  acted  alone  if 
the  fluid  had  been  immovable. 

602.  From  the  foregoing  principles  it  will  be  easy  to  under- 
stand how  induction  may  operate  through  a  succession  of  con- 
ductors, which  are  all  of  them  insulated  except  the  last ;  and 
which  are  separated  from  each  other  by  distances  greater  than 
that  at  which  a  transfer  of  electricity  would  take  place.     If,  un- 
der such  circumstances,  the  first  be  electrified,  alternate  states 
of  opposite  electricities  will  be  produced  in  the  two  ends  of  each 
conductor  in  succession.     In  all  the  en.ds  nearest  the  first  body, 
the  electricity  will  be  of  the  opposite  kind  to  that  with  which 
the  first  has  been  charged  ;  in  the  other  ends  it  will  be  of  the 
same  kind  as  that  of  the  first  body.     The  vicinity  of  these  op- 
posite electricities  will  tend  powerfully  to  retain  them  in  that 
condition,  and  will  diminish  their  electric  action  on  surrounding 
bodies.     A  large  portion  of  the  electricities  so  arranged  and  re- 
tained, is  therefore  in  the  condition  designated  by  the  term  dis- 
guised electricity.*     (Art.  599.) 

The  principles  of  induction  developed  in  the  preceding  arti- 
cles, serve  to  explain  a  number  of  the  most  curious  and  intricate 
phenomena  of  electricity,  among  which  are  those  of  the  Leyden 
jar ;  to  this  instrument,  therefore,  let  us  now  return. 

*  Lib.  of  Use.  Knowl.,  Art.  Electricity 


412  NATURAL   PHILOSOPHY. 


THEORY    OF    THE    LEYDEN    JAR. 

603.  Upon  what  principle  does  this  instrument  receive  and 
retain  such  an  accumulation  of  the  electric  fluid  ?     The  answer 
is,  because  the  two  surfaces  of  the  jar  mutually  augment  each  other's 
capacities,  upon  the  principle  of  induction.     To  trace  the  opera- 
tion of  this  principle  a  little  more  particularly,  let  us  observe 
what  takes  place  while  a  jar  is  charging  from  the  prime  con- 
ductor of  the  electrical  machine.     And  first,  suppose  the  jar  is 
insulated  ;  a  spark  passes  to  the  inner  surface,  and  electrifies  it 
positively.     The  inner  surface  now  stands  in  the  same  relation 
to  the  outer,  that  the  globe  in  Fig.  221  stands  to  the  cylinder ; 
that  is,  it  tends  to  drive  off  the  electricity  of  the  same  kind,  and, 
in  the  same  proportion,  to  attract  the  electricity  of  the  opposite 
kind.     But  as  the  fluid  cannot  escape  from  the  outer  surface, 
(the  jar  being  insulated,)  it  of  course  remains  to  oppose  the  fur- 
ther accumulation  of  the  similar  fluid  on  the  inner  surface.    But 
secondly,  suppose  the  jar  uninsulated,  its  outer  coating  having  free 
communication  with  the  earth.     A  spark  passes  to  the  inside  as 
before,  and  electrifies  positively  the  inner  coating.     This  repels 
the  similar  electricity  from  the  outer  coating,  and  renders  the 
outside  negative.     Being  negative  it  reacts  by  induction,  (as  the 
nearer  surface  of  the  cylinder,  in  Art.  595,)  on  the  inside,  and 
attracts  to  it  a  still  greater  charge,  which  is  supplied  by  the 
prime  conductor.     This  additional  charge,  acting  in  the  same 
manner  on  the  outside,  renders  it  more  highly  negative  than  be- 
fore, in  consequence  of  which  it  attracts  to  the  inside  a  still  fur- 
ther charge  of  electricity  from  the  machine.     This  series  of  ac- 
tions and  reactions  between  the  two  surfaces  of  the  jar,  proceeds 
in  a  diminishing  series,  until  each  surface  becomes  too  feeble  to 
exert  any  further  influence  on  the  other,  and  the  jar  is  then 
charged. 

Substituting  the  terms  vitreous  and  resinous,  for  positive  and 
negative,  as  in  Art.  600,  we  may  easily  make  the  foregoing  ex- 
planation conform  to  the  supposition  of  two  fluids. 

604.  For  the  purpose  of  making  the  theory  of  the  Leyden  jar 
familiar,  we  may  now  recur  to  the  experiments  mentioned  in 
Art.  594,  and  attempt  the  explanation  of  them. 

In  the  structure  of  the  jar,  we  recognise  the  operation  of  the 
principle  of  induction.  Here,  an  unelectrified  body  (the  outer 
surface)  is  brought  very  near  to  an  electrified  body,  (the  inner 
surface,)  without  the  possibility  of  communicating  with  each 
other,  on  account  of  the  non-conducting  properties  of  the  glass. 
The  nearer  the  two  surfaces  can  be  brought  to  each  other,  the 
more  powerful  is  the  effect  of  induction,  that  effect  being  in- 
versely as  the  square  of  the  distance.  Accordingly,  the  thinner 
the  jar,  the  more  powerful  is  the  charge  it  will  receive  ;  but  the 


ELECTRICITY.  413 

danger  of  breaking  prevents  our  employing  such  as  are  very 
thin.* 

To  trace  the  process  of  charging  a  jar  a  little  more  minutely, 
let  us  suppose  the  jar  connected  with  the  prime  conductor  of  an 
electrical  machine,  from  which  a  spark  is  communicated  to  the 
inner  coating.  This,  according  to  the  principle  of  induction, 
expels  a  similar  quantity  of  the  same  fluid  from  the  opposite  un- 
electrified  surface,  and  renders  that  negative,  in  the  same  de- 
gree as  the  inside  is  positive.  Being  negative,  it  increases  the 
attraction  of  the  inner  surface  for  the  opposite  species  of  fluid, 
and  another  spark  is  received,  which  again  expels  an  additional 
quantity  of  the  same  species  of  fluid  from  the  outside,  and  thus 
the  two  surfaces  continue  to  act  upon  each  other  reciprocally, 
though  with  constantly  diminishing  power,  until  the  jar  is 
charged. 

The  reason  also  is  plain,  why  the  outside  of  the  jar  must  be 
uninsulated  ;  since  it  is  only  in  such  case,  that  the  foregoing 
process  of  induction  can  take  place  ;  and  we  readily  see  why  a 
series  of  jars  may  be  charged,  from  the  portion  of  electricity 
which  is  expelled  from  the  outside  of  the  first  jar. 

605.  When  a  jar  is  charged  negatively  from  the  rubber,  just 
the  opposite  process  in  all  respects  takes  place,  the  outside  be- 
coming  positive  by  induction,  and  reacting  upon   the   inside. 
The  case  mentioned  in  Art.  594,  (6,)  where  two  jars  differently 
charged,  cannot  be  discharged  unless  their  outer  surfaces  be  in 
conducting  communication,  will  be  readily  understood  ;  for  it  is 
impossible  for  the  equilibrium  to  be  restored  by  the  union  of  the 
electricities  on  the  inside,  while  the  outside  remains  electrified. 
If  we  could  suppose  this  to  take  place  for  a  moment,  and  the 
electricity  within  to  be  restored  to  its  natural  state,  it  would 
again  be  immediately  decomposed  by  the  inductive  influence  of 
the  electrified  coating  without. 

606.  The  phenomena  of  the  Leyden  jar,  may  be  equally  well 
explained,  by  substituting  the  terms  vitreous  and  resinous,  in- 
stead of  positive  and  negative,  on  the  supposition  of  two  fluids, 
since  the  principles  of  induction  apply  equally  well  to  both  hy- 
potheses.    Thus,  it  is  as  easy  to  suppose  that  the  resinous  electri- 
city is  induced  upon  the  outside  by  the  attraction  of  the  vitreous 
electricity  within,  as  it  is  to  suppose  that  the  outside  becomes 
negative  by  the  loss  of  a  portion  of  its  natural  share  ;  and  the 
necessity  of  the  outer  surface  being  uninsulated,  is  as  apparent 
in  the  one  case  as  in  the  other.     But  we  reserve  the  discussion 
of  the  comparative  merits  of  these  remarkable  hypotheses,  un- 

*  The  writer  of  this  Treatise  had  a  large  jar  constructed  of  very  thin  glass ;  it  took 
an  extraordinary  charge,  and  when  discharged  gave  a  report  like  that  of  an  ordinary 
battery ;  but  it  was  fractured  by  the  first  experiment. 


414  NATURAL   PHILOSOPHY. 

til  the  learner  shall  have  become  familiar  with  a  great  variety 
of  electrical  phenomena. 


CHAPTER  IV. 

OF  ELECTRICAL  LIGHT. 

607.  LIGHT,  we  have  seen,  is  not  a  constant  attendant  of  elec- 
trical phenomena.     Indeed,  until  noticed  by  Otto  Guericke,  it 
it  was  not  known  to  have  any  relation  to  electricity. 

Electrical  light  appears  whenever  the  fluid  is  discharged,  in  con- 
siderable quantity,  through  a  resisting  medium. 

Accordingly,  no  light  is  perceived  when  electricity  flows  freely 
through  good  conductors  ;  but  if  such  conductors  suffer  any  in- 
terruption, as  by  the  intervention  of  a  space  of  air,  or  even  of  an 
imperfect  conductor,  then  the  attendant  light  becomes  manifest. 

608.  We  shall  best  learn  the  properties  of  the  electrical  spark, 
by  attending  to  a  variety  of  experiments  in  which  it  is  exhibited.* 

A  glass  tube  rubbed  with  black  silk,  which  has  been  smeared  with 
a  little  electrical  amalgam,  will  yield  copious  sparks,  and  flashes 
of  light.  The  tube  should  be  warm,  dry,  and  smooth,  and  of  a 
size  not  less  than  two  feet  in  length,  and  three  fourths  of  an  inch 
in  diameter. 

The  electrical  machine,  when  in  vigorous  action,  affords  brilliant 
circles  and  streams  of  light.  In  order  to  render  the  light  afforded 
by  turning  the  machine  abundant,  several  practical  expedients 
are  necessary.  All  parts  of  the  machine  must  be  dry  and  warm, 
(but  not  hot.)  It  is  useful  to  rub  very  freely  the  glass  plate  or 
cylinder,  with  an  old  silk  handkerchief.  Black  spots,  or  lines 
that  collect  on  the  glass,  especially  when  the  amalgam  is  new, 
are  to  be  carefully  rubbed  off,  and  should  dust  or  down  collect 
on  the  amalgam  of  the  rubber,  this  must  be  removed.  The  ac- 
tion of  the  cylinder  will  be  increased  by  the  following  process : 
rub  a  little  tallow  on  the  palm  of  the  hand,  an'd  apply  it  to  the 
bottom  of  the  cylinder  ;  then  turn  the  machine  until  the  tallow 
is  all  taken  up  by  the  rubber  and  flap.  The  pores  of  the  flap 
will  then  become  filled  with  tallow,  it  will  apply  itself  more  close- 
ly to  the  cylinder,  and  the  supply  of  electricity  will  become  more 
copious.  A  convenient  method  of  recruiting  the  action  of  the 
machine,  is  to  coat  a  circular  disk  of  pasteboard  or  leather  with 

*  In  experiments  on  electrical  light,  the  room  is  supposed  to  be  dark.  They  ap- 
pear to  the  best  advantage  in  the  night. 


ELECTRICITY.  415 

amalgam,  and  to  apply  it  to  the  glass  plate  or  cylinder  while  the 
machine  is  turning. 

If  the  chain  be  removed  from  the  rubber  to  the  prime  conduct- 
or, so  that  the  former  shall  be  insulated  and  the  latter  uninsu- 
lated, on  bringing  the  ends  of  the  fingers  near  the  rubber,  a 
stream  of  diluted  light  will  pass  between  the  fingers  and  the 

rubber. 

/ 

609.  The  length,  color,  and  form  of  the  electric  spark,  varies 
.with  the  nature  of  the  conductors  between  which  it  passes,  and  with 

that  of  the  medium  interposed  between  them. 

Electrical  sparks  are  more  brilliant  in  proportion  as  the  sub 
stances  between  which  they  occur  are  better  conductors.  A 
spark  received  from  the  prime  conductor  upon  a  large  metallic 
ball,  is  short,  straight,  and  white ;  on  a  small  ball  it  is  longer, 
and  zigzag;  received  on  the  knuckle,  a  less  perfect  conductor, 
it  is  purplish  or  reddish  ;  on  wood,  or  ice,  or  a  wet  plant,  or  wa- 
ter, it  is  red.  Moreover,  a  longer  spark  can  be  obtained  from  a 
small  ball,  attached  to  the  prime  conductor  by  a  wire  of  five  or 
six  inches  long,  than  from  the  prime  conductor  itself;  and  the 
longest  and  most  zigzag  spark  is  obtained,  when  the  knob  of  a 
Leyden  jar  is  presented  to  a  similar  brass  ball  attached  to  the 
prime  conductor.  From  a  point  positively  electrified,  the  fluid 
passes  in  the  form  of  a  brush  or  pencil  of  rays ;  a  point  connect- 
ed with  the  negative  side,  exhibits  a  luminous  star. 

A  metallic  chain  connected  with  the  prime  conductor,  be- 
comes illuminated  at  the  points  where  two  links  join,  and  at  oth- 
er points  where  the  conducting  powers  of  the  metal  are  impair- 
ed by  rust,  or  where  roughnesses  occur.  If  the  chain  has  been 
previously  corroded,  artificially,  by  dipping  it  into  a  solution  of 
salt,  or  a  strong  acid,  and  suffering  it  to  remain  until  the  outside 
has  become  rusty,  the  experiment  will  be  more  striking.  When 
the  chain  is  so  good  a  conductor  as  to  afford  a  ready  passage  to 
the  fluid,  the  light  will  be  produced  more  abundantly  if  the  re- 
moter end  of  the  chain  be  held  by  the  discharging  rod,  so  as  to 
insulate  it ;  or  it  may  be  attached  to  any  other  insulating  sup- 
port. 

610.  The  electric  spark  passes,  with  increased  facility,  through 
rarefied  air ;  and  the  distance  to  which  it  will  pass  between  two 
conductors,  is  augmented  as  the  rarefaction  is  made  more  complete. 

Instead  of  the  distance  of  five  or  six  inches,  which  is  the  limit 
of  the  spark  from  the  prime  conductor  of  an  ordinary  machine 
in  the  open  air,  the  spark  will  pass  through  the  space  of  four 
feet  or  more,  in  an  exhausted  tube.  If  a  pointed  wire,  termina- 
ting in  a  knob  above,  be  introduced  into  the  top  of  a  tall  re- 
ceiver, and  the  receiver  be  placed  on  the  plate  of  the  air-pump, 


416  NATURAL   PHILOSOPHY. 

on  connecting  the  knob  of  the  wire  Avith  the  prime  conductor 
and  turning  the  machine,  a  brush  of  light  only  will  appear  at 
the  extremity  of  the  wire;  but,  on  exhausting  the  air,  this  brush 
will  enlarge,  varying  its  appearance,  and  becoming  more  dif- 
fused as  the  air  becomes  more  rarefied,  until  at  length  the  whole 
receiver  is  pervaded  by  a  beautiful  bluish  light,  changing  its  co- 
lor with  the  intensity  of  the  transmitted  electricity,  and  produ- 
cing an  effect  which,  with  an  air-pump  of  considerable  power, 
is  pleasing  in  the  highest  degree.* 

When  a  charged  jar  is  placed  under  the  receiver  of  an  air- 
pump,  as  the  exhaustion  proceeds,  a  luminous  current  flows  over 
the  edge  of  the  jar  from  the  positive  to  the  negative  side,  until 
the  equilibrium  is  restored,  f 

611.  Electric  light  exhibits  a  very  beautiful  appearance  as  it 
passes  or  flows  through  the  Torricellian  Vacuum.     The  color  is 
of  a  very  delicate  bluish  or  purple  tinge,  and  the  light  pervades 
the  entire  space.     But  the  most  pleasing  exhibitions  of  this  kind 
are  made  by  forming  an  artificial  atmosphere  of  vapor  in  the 
Torricellian  tube.     Ether,  or  alcohol,  passes  into  the  state  of  va- 
por when  the  pressure  of  the  atmosphere  is  removed  ;  and,  ac- 
cordingly, on  introducing  a  drop  of  one  of  these  fluids  into  the 
Torricellian  vacuum,  it  immediately  evaporates  and  fills  the 
void.     If,  now,  a  strong  spark  be  passed  from  the  prime  condjict- 
or  through  this  vapor,  the  spark  will  exhibit  various  colors ;  in 
ether,  it  is  an  emerald  green,  or  mingled  red  and  green  ;  in  alco- 
hol, it  is  red  or  blue  ;  but  the  colors  vary  somewhat  with  the  dis- 
tances at  which  they  are  seen.J 

612.  Sir  Humphry  Davy  performed  a  number  of  experiments, 
on  the  passage  of  electricity  through  a  vacuum,  of  which  an  ac- 
count is  given  in  the  Philosophical  Transactions  for  1822.     He 
succeeded  in  forming  a  Torricellian  vacuum  quite  free  from  air, 
but  in  such  cases,  a  small  portion  of  the  mercury  itself  is  con- 
verted into  vapor,  and  from  this  he  could  not  free  the  empty 
space.     In  all  cases  where  the  mercurial  vacuum  was  perfectly 
free  from  air,  it  was  permeable  to  electricity,  and  was  rendered 
luminous  by  either  the  common  spark,  or  the  discharge  from  a 
Leyden  jar.     But  the  degree  of  the  intensity  of  this  phenomenon 
depended  upon  the  temperature.     When  the  tube  was  very  hot, 
the  electric  light  appeared  in  the  vapor  of  a  bright  green  color, 
and  of  great  density.     As  the  temperature  diminished,  it  lost  its 
vividness,  and  when  it  was  artificially  cooled  to  20°  below  zero, 
it  was  so  faint  as  to  be  visible  only  in  the  dark.     In  all  cases, 
where  the  minutest  quantity  of  rarefied  air  was  introduced  into 

*  Singer.  t  Ib. 

t  A  Torricellian  tube  may  be  prepared  for  this  purpose,  by  closing  the  upper  end, 
next  the  vacuum,  with  a  cork,  having  a  large  pin  passing  through  it,  the  size  of  the 
pin-head  being  enlarged  by  a  small  ball  of  pith  or  cork. 


ELECTRICITY.  41? 

the  mercurial  vacuum,  the  electrical  light  changed  from  green 
to  sea  green,  and  by  increasing  the  quantity  of  air,  it  changed  to 
blue  and  purple.  Also  when  the  temperature  was  low,  the  vac- 
uum became  a  much  better  conductor. 

A  more  perfect  vacuum  was  formed  by  means  of  melted  met- 
als, as  tin,  of  a  more  fixed  nature  than  mercury,  and  therefore 
not  liable  to  impair  the  vacuum  by  vapor  of  their  own.  A  vacu- 
um being  made  by  means  of  fused  tin,  the  electric  light,  at  tem- 
peratures below  zero,  was  yellow,  and  of  the  palest  phosphores- 
cent kind,  requiring  almost  absolute  darkness  to  be  perceived  ; 
nor  was  it  perceptibly  increased  by  heat.  When  the  tempera- 
ture was  diminished,  the  electrical  light  (transmitted  through 
vapor  of  mercury)  diminished  also,  till  the  temperature  was  re- 
duced to  20°  ;  but  between  20°  and  —20°  it  seemed  stationary. 

Unless  the  electrical  machine  was  very  active,  no  light  was 
visible  during  the  transmission  of  electricity  $  but  that  the  elec- 
tricity passed  was  evident,  from  the  luminous  appearance  of  the 
rarefied  air,  in  other  parts  of  the  tube. 

From  these  and  various  similar  experiments  related  by  Davy, 
it  seems  demonstrated,  that  electricity  is  capable  of  passing 
through  a  perfect  vacuum,  but  that  the  light  emitted  depends 
upon  the  vapor  or  air  through  which  it  passes,  and  that  if  the 
vacuum  were  perfect,  no  light  whatever  would  appear.* 

In  condensed  air,  on  the  contrary,  the  spark  passes  with  great- 
er difficulty  than  ordinary.  In  such  case,  also,  its  whiteness  and 
brilliancy  are  augmented,  and  its  course  is  zigzag.  These  ap- 
pearances are  even  exhibited  by  passing  the  spark  through  con- 
fined air,  of  only  the  ordinary  density. 

613.  The  colors  of  the  spark  are  pleasingly  varied  by  passing 
it,  in  a  condensed  form,  as  in  the  Leyden  jar,  through  media  of 
different  kinds.  The  experiment  is  performed  by  making  the 
given  body  form  a  part  of  the  circuit  of  communication,  between 
the  inside  and  outside  of  the  Leyden  jar.  A  ball  of  ivory  in  this 
situation  exhibits  a  beautiful  crimson ;  an  egg,  a  similar  color, 
but  somewhat  lighter ;  a  lump  of  sugar  gives  a  very  white  light, 
which  remains  for  some  time  after  the  spark  has  passed ;  and 
fluor  spar  exhibits  an  emerald  green  light,  or,  in  some  cases, 
a  purple  light,  which  also  continues  to  glow  in  the  dark  for  some 
seconds.  The  great  intensity  of  the  light  is  shown  by  the  strong 
illumination  which  the  sparks  in  the  jar  communicate  to  bodies 
slightly  transparent.  Thus  an  egg  has  its  transparency  greatly 
increased ;  and  if  the  thumb  be  placed  over  the  space  which 
separates  the  two  conducting  wires  that  communicate  with  the 
two  sides  of  the  jar  respectively,  the  illumination  is  so  powerful, 
that  the  blood-vessels  and  interior  structure  of  the  organ  may 
be  distinctly  seen. 

*  Phil  Trans.  1822,  or  Thomson,  Outlines,  p.  470. 
53 


418 


NATURAL   PHILOSOPHY. 


614.  Metallic  conductors,  if  of  sufficient  size,  transmit  elec- 
tricity without  any  luminous  appearance,  provided  they  are  per- 
fectly continuous ;  but  if  they  are  separated  in  the  slightest  de- 
gree, a  spark  will  occur  at  every  separation.  On  this  principle, 
various  devices  are  formed,  by  pasting  a  narrow  band  of  tinfoil 
on  glass,  in  the  required  form,  and  cutting  it  across  with  a  pen- 
knife, where  we  wish  sparks  to  appear.  If  an  interrupted  con- 
ductor of  this  kind  be  pasted  round  a  glass  tube  in  a  spiral  di- 
rection, and  one  end  of  the  tube  be  held  in  the  hand,  and  the 
other  be  presented  to  an  electrified  conductor,  a  brilliant  line  of 
light  surrounds  the  tube,  which  has  been  called  the  spiral  tube, 
or  diamond  necklace.  By  inclosing  the  spiral  tube  in  a  larger 
cylinder  of  colored  glass,  the  sapphire,  topaz,  emerald,  and  other 
gems  may  be  imitated.  Words,  flowers,  and  other  complicated 
forms,  are  also  produced  nearly  in  the  same  manner,  by  a  proper 
disposition  of  an  interrupted  line  of  metal,  on  a  flat  piece  of 


615.  The  light  of  the  electric  spark,  is  not  a  constituent  part  of 
the  electricity,  but  arises  from  the  sudden  compression  of  the  air,  or 
other  medium  through  which  it  passes. 

It  is  well  known,  that  air  is  capable  of  affording  a  spark  by 
sudden  compression.  There  is  a  kind  of  match  constructed  on 
this  principle,  in  which  a  small  portion  of  air  contained  in  a  close 
cylinder,  being  suddenly  compressed  by  forcing  down  a  piston, 
yields  a  spark  sufficient  to  light  a  quantity  of  tinder  at  the  bot- 
tom of  the  cylinder.  Now  it  is  found  by  actual  experiment,  that 
electricity  has  the  power  of  condensing  air.  This  fact  is  shown 
by  means  of  a  small  instrument  called  Kinnersleifs  Air  Ther- 
mometer. It  consists  of  a  glass  tube,  closed  air-tight  Fig.  222. 
at  the  two  ends  by  brass  caps,  through  each  of  which 
passes  a  movable  wire,  terminated  within  by  a  small 
ball.  Through  the  lower  cap  is  inserted  a  small  glass 
tube,  open  at  both  extremities,  and  turned  upward 
parallel  to  the  cylinder.  Into  this  tube  is  introduced 
a  quantity  of  water  sufficient  to  cover  the  bottom  of 
the  cylinder,  and  of  course  to  rise  a  little  way  into 
the  tube.  The  two  balls  being  set  at  some  distance 
from  each  other,  and  a  spark  from  the  Leyden  jar 
being  passed  between  them,  the  air  within  is  sudden- 
ly rarefied,  and  the  water  ascends  in  the  tube,  and 
again  descends,  when  the  explosion  is  over.  This 
sudden  rarefaction  of  a  portion  of  air  before  the  elec- 
tric spark,  must  cause  a  sudden  and  powerful  com- 
pression in  the  portions  of  air  immediately  adjacent. 
The  immense  velocity  of  the  spark  must  greatly 
increase  the  resistance,  and  of  course  the  force  of  compression. 
This  appears  to  be  an  adequate  cause  for  the  production  of  the 


ELECTRICITY.  419 

light  that  accompanies  the  electric  discharge,  and  hence  we 
conclude,  that  light  is  not  inherent  in  the  fluid  itself.  The 
greater  density  and  brilliancy  of  the'  spark  in  condensed  air, 
and  its  feebleness  and  diffuseness  in  a  rarefied  medium,  are 
facts  which  accord  well  with  the  supposed  origin  ;  and  the  zig- 
zag form  of  the  spark  when  long,  or  when  passing  through  con- 
densed air,  is  well  explained  by  the  same  theory.  For  the  elec- 
tric fluid  in  its  passage  through  the  air,  condenses  the  air  before 
it,  and  thus  meets  with  a  resistance  which  turns  it  off  laterally ; 
in  this  direction  it  is  again  condensed,  and  has  its  course  again 
changed  ;  and  so  on,  until  it  reaches  the  conductor  toward  which 
it  is  aiming.  The  zigzag  form  of  lightning  is  accounted  for  on 
this  principle.* 

Electrical  light  is  found,  by  optical  experiments,  to  have  in 
general  the  same  nature  with  the  light  of  the  sun, — being  like 
this  resolved  into  various  colors  by  the  prism,  and  possessing 
other  properties,  to  be  described  under  the  head  of  Optics,  which 
identify  it  with  solar  light. 


CHAPTER  V. 

OF  THE  ELECTRIC  BATTERY.  —  MECHANICAL  AND  CHEMICAL 
AGENCIES,  AND  MOTIONS  OF  ELECTRICITY.  —  EFFECTS  OF 
ELECTRICITY  ON  ANIMALS. 

616.  An  electric  battery  consists  of  a  number  of  Ley  den  jars  so 
combined,  that  the  whole  may  be  either  charged  or  discharged  at 
once. 

Very  large  jars  cannot  be  obtained :  it  is  rare  to  find  one  more 
than  two  feet  high,  by  one  and  a  half  in  diameter.  Yet  some 
of  the  mechanical  effects  of  electricity  to  be  described  hereafter, 
require  a  much  greater  accumulation  of  the  fluid  than  can  be 
obtained  from  any  single  jar.  The  battery  is  constructed  as 
follows.  Large  jars,  twelve  or  fourteen  inches  high  by  five  or 
six  inches  in  diameter,  are  coated  like  ordinary  Leyden  jars. 
Twelve  of  these  constitute  a  battery  sufficiently  powerful  for 
most  purposes,  but  the  power  of  the  battery  may  be  carried  to 
an  indefinite  extent  by  increasing  the  number  of  jars.  When  the 
number  is  twelve,  they  are  placed  four  in  a  row  in  a  box,  the 
bottom  of  which  is  coated  with  tinfoil,  by  means  of  which  the 
outsides  of  the  jars  are  all  in  conducting  communication.  Each 
jar  is  separated  from  the  rest  by  a  slight  partition  of  wood.  To 
connect  the  insides  of  the  jars,  their  knobs  are  joined  by  large 

»  Biot,  Traite  de  Phys.,  tome  2.— Encyc.  Metropol.,  Art.  Electricity. 


420  NATURAL   PHILOSOPHY. 

brass  wires.  It  is  obvious,  therefore,  that  the  battery  is  equiva- 
lent to  a  single  jar  of  enormous  size,  comprehending  the  same 
number  of  square  feet. 

The  object  of  the  battery  is  to  accumulate  a  great  quantity  of 
the  electric  fluid,  which  is  in  proportion  to  the  extent  of  surface : 
the  intensity  or  elastic  force,  as  indicated  by  the  quadrant  elec- 
trometer, is  no  greater  in  the  battery  when  charged,  than  in  a 
single  charged  jar.  The  battery,  like  the  common  jar,  is  charged 
by  bringing  the  inside  into  communication  with  the  prime  con- 
ductor of  an  active  and  powerful  electrical  machine  :*  it  is  dis- 
charged as  usual,  by  forming  a  connection  between  the  inside 
and  outside,  commonly  by  means  of  the  discharging  rod. 

Electrical  batteries  indicate  only  the  intensity  of  the  accumu- 
lated electricity,  that  is,  its  deviation  from  a  state  of  natural 
distribution ;  the  quantity  can  be  inferred  only  from  the  com- 
parative extent  of  the  charged  surface,  or  estimated  by  an  ex- 
amination of  its  effects,  and  is  therefore  by  no  means  accurately 
appreciable. 

617.  The  largest  machine  and  battery  hitherto  constructed, 
were  made  for  the  Teylerian  museum,  at  Haarlem.  It  consists 
of  two  circular  plates  of  glass,  each  five  feet  five  inches  in  di- 
ameter. The  prime  conductor  consists  of  several  pieces,  and  is 
supported  by  three  glass  pillars,  nearly  five  feet  in  length.  The 
force  of  two  men  is  required  to  work  the  machine  ;  and  when 
it  is  required  to  be  put  in  action  for  any  length  of  time,  four  are 
necessary.f 

At  its  first  construction  nine  batteries  were  applied  to  it,  each 
having  fifteen  jars,  every  one  of  which  contained  a  square  foot 
of  coated  glass  ;  so  that  the  grand  battery  formed  by  the  combina- 
tion of  all  these,  contained  one  hundred  and  thirty-five  feet.  As 
examples  of  the  great  power  of  the  Teylerian  machine,  we  may 
mention  the  following  :  it  charged  a  Leyden  jar  by  turning  the 
handle  half  round, — a  charge  which  the  jar  would  receive  and 
lose  by  discharging  itself  spontaneously  eighty  times  in  a  minute. 
A  single  spark  from  the  conductor  melted  a  considerable  length 

*  As  the  process  of  charging  a  large  battery  is  tedious  and  laborious,  it  has  been 
proposed  to  charge  each  jar  successively,  after  that  which  is  immediately  connected 
to  the  prime  conductor,  by  means  of  the  electricity  expelled  from  the  outside  of  the 
first,  as  is  explained  in  article  594,  (4.) — Encyc.  Metrop.,  Art.  Electricity. 

t  Wright's  Electrical  Machine  is  one  of  the  largest  hitherto  constructed  in  this 
country.  It  has  four  large  cylinders,  all  made  to  turn  simultaneously,  and  to  deliver 
their  electricity  to  the  same  prime  conductor.  The  prime  conductor  is  made  of  sheet 
brass,  of  which  it  contains  more  than  thirty  square  feet,  and  is  mounted  horizontally 
on  glass  pillars  immediately  over  the  cylinders.  The  spark  from  this  conductor,  when 
all  the  cylinders  are  in  full  action,  received  on  the  knuckle,  gives  a  painful  sensation. 
The  machine  charges  a  battery  consisting  of  twelve  gallon  jars  in  thirty  seconds,  and 
causes  a  half  gallon  jar  to  explode  spontaneously  twelve  times  in  a  minute.  T 
elegant  piece  of  apparatus  was  constructed  by  Mr.  Caleb  Wright,  and  has  recently 
been  presented  by  that  gentleman  to  Yale  College. 


ELECTRICITY.  421 

of  gold  leaf.  A  spark  or  zigzag  stream  of  fire  would  dart  from 
the  prime  conductor  to  a  neighboring  conductor,  to  the  distance 
of  ten  feet.  A  wire  three  eighths  of  an  inch  in  diameter  was 
found  to  be  insufficient  to  transmit  the  whole  charge  of  the 
prime  conductor,  but  the  wire,  vmald  give  small  sparks  to  a  con- 
ductor brought  near  it.  The  sphere  of  influence  (Art.  575)  ex- 
tended to  the  distance  of  forty  feet,  so  as  sensibly  to  affect  the 
pith  ball  electrometer.  The  spider  web  sensation,  (or  that  pecu- 
liar sensation  resembling  that  of  the  spider's  web,)  which  is  ex- 
perienced by  holding  an  excited  glass  tube  to  the  face,  was  felt 
by  bystanders  to  a  great  distance  from  the  machine.* 


MECHANICAL   EFFECTS   OF   ELECTRICITY. 

618.  The  sound  produced  by  an  electric  discharge  is  ascribed  to 
the  sudden  collapse  of  the  air,  which  has  been  displaced  by  the  pas 
sage  of  the  electric  fluid. 

Hence  the  sound  is  greater  in  proportion  to  the  quantity  and 
intensity  of  the  charge.  A  battery,  when  fully  charged,  gives  a 
loud  explosion. 


619.  Imperfectly  conducting 
ful  electric  charge  is  passed,  ai 


substances,  through  which  a  power - 

^^  ,  are  torn  asunder  with  more  or  less 

violence. 

A  large  Ley  den  jar  is  sufficient  for  exhibiting  some  of  these 
mechanical  effects  :  others  require  the  power  of  the  battery. 
When  the  charge  is  passed  through  a  thick  card,  or  the  cover  of 
a  book,  a  hole  is  torn  through  it,  which  presents  the  rough  ap- 
pearance of  a  bur  on  each  side.  By  means  of  the  battery,  a 
quire  of  strong  paper  may  be  perforated  in  the  same  manner ; 
and  such  is  the  velocity  with  which  the  fluid  moves,  that  if  the 
paper  be  freely  suspended,  not  the  least  motion  is  communicated 
to  it.f  (See  Art.  231.)  Pieces  of  hard  wood,  of  loaf  sugar,  of 
stones,  and  many  other  brittle  non-conductors,  are  broken,  or 
even  torn  asunder  with  violence,  by  a  powerful  charge  from  the 
battery. 

The  expansion  of  fluids  by  electricity  is  very  remarkable,  and 
productive  of  some  singular  results.  When  the  charge  is  strong, 
no  glass  vessel  can  resist  the  sudden  impulse.  Beccaria  inserted 
a  drop  of  water  between  two  wires  in  the  center  of  a  solid  glass 
ball  of  two  inches  diameter :  on  passing  a  shock  through  the 
drop  of  water,  the  ball  was  dispersed  with  great  violence.  In 
like  manner,  by  the  sudden  expansion  of  a  small  body  of  confined 
air,  strongly  electrified,  explosions  may  be  produced,  and  bodies 
that  resist  its  expansion  are  projected  with  violence.  Even  good 
conductors,  when  minutely  divided,  are  expanded  by  electricity. 

*  Cavallo,  Complete  Treatise,  Vol.  II.  t  Singer. 


422  NATURAL   PHILOSOPHY. 

Thus  mercury,  confined  in  a  capillary  glass  tube,  will  be  ex- 
panded with  a  force  sufficient  to  splinter  the  tube.* 

CHEMICAL  EFFECTS  OF  ELECTRICITY. 

620.  By  means  of  electricity,  more  or  less  accumulated,  a  variety 
of  chemical  effects  may  be  produced  ;  such  as  the  combustion  of  in- 
flammable bodies,  the  oxidation,  fusion,  and  even  combustion  of 
metals,  the  separation  of  compounds  into  their  elements,  or  the  union 
of  elements  into  compounds.-^ 

Ether  and  alcohol  may  be  inflamed  by  passing  the  electric 
spark  through  them ;  nor  is  the  effect  diminished  by  communi- 
cating the  spark  by  means  of  a  piece  of  ice  or  any  other  cold 
medium.  The  finger  may  be  conveniently  employed  to  inflame 
these  substances.  Phosphorus,  resin,  and  other  solid  combusti- 
ble bodies,  may  be  set  on  fire  by  the  same  means  ;  gunpowder 
and  the  fulminating  powders  may  be  exploded,  and  a  candle 
may  be  lighted.  Gold  leaf  and  fine  iron  wire  may  be  burned, 
by  a  charge  from  the  battery.  Wires  of  lead,  tin,  zinc,  copper, 
platina,  silver,  and  gold,  when  subjected  to  the  charge  of  a  very 
large  battery,  burn  with  explosion,  and  are  converted  into  ox- 
ides.J 

The  same  agent,  moreover,  is  capable  of  reviving  these  oxides ; 
that  is,  restoring  them  to  the  state  of  pure  metals.  By  a  singu- 
lar contrariety  of  properties,  water  is  decomposed  into  its  gase- 
ous elements,  and  the  same  elements  are  reunited  to  form  water ; 
and  the  constituent  gases  of  atmospheric  air,  are,  by  passing  a 
great  number  of  electric  charges  through  a  confined  portion  of 
air,  converted  into  nitric  acid. 

621.  The  velocity  of  the  electric  jluid  is  apparently  instantane- 
ous.    A  circuit  of  four  miles  has  been  formed,  by  means  of  wire, 
between  the  inside  and  outside  of  a  Leyden  jar,  and  no  percepti- 
ble interval  was  occupied  during  the  discharge.     Analogy,  how- 
ever, would  lead  us  to  believe  that  electricity,  like  light,  is  pro- 
gressive in.  its  motions,  but  that  it  moves  with  a  velocity  too 
great  to  be  measured,  except  for  intervals  of  immense  extent. 

The  velocity  of  light  appears  to  be  instantaneous,  for  such 
distances  as  four  miles ;  but  when  such  intervals  are  taken  as 
the  diameter  of  the  earth's  orbit,  light  is  found  to  have  a  pro- 

*  The  illuminating  powers  of  the  spark  from  a  large  battery,  produce  very  striking 
appearances.  The  scintillations  of  an  iron  chain  20  or  30  feet  long,  when  made  to 
form  the  medium  of  communication  between  the  inside  and  outside  pf  a  charged  bat- 
tery, are  peculiarly  beautiful.  The  chain  may  conveniently  be  hung  in  catenary 
curves  from  the  ceiling  of  the  room,  supported  by  threads  of  silk. 

t  The  chemical  agencies  of  electricity,  however,  are  much  more  powerful  and  ex- 
tensive as  exhibited  by  the  galvanic  apparatus,  than  by  the  common  electrical  ma- 
chine. 

t  Singer,  p.  185. 


ELECTRICITY.  423 

gressive  velocity  of  192,500  miles  per  second.  If,  therefore,  elec- 
tricity actually  moves  with  a  progressive  velocity  like  that  of 
light,  still  the  time  occupied  in  traversing  the  space  of  four  miles 
would  be  inappreciable,  since  it  would  equal  only  about  the  fifty 
thousandth  part  of  a  second.  Experiments  are  in  progress  for 
conveying  telegraphic  dispatches  by  means  of  electricity. 

622.  The  electric  fluid,  in  its  route,  selects  the  best  conductors. 
The  Leyden  jar  may  be  discharged  with  a  wire  held  in  the  hand, 
without  the  insulating  handle  used  in  the  discharging  rod,  since 
metallic  wire  is  a  better  conductor  than  the  hand,  and  the  fluid 
will  take  its  route  through  that  in  preference  to  the  hand.     But 
if  a  wooden  discharger  be  substituted  for  the  wire,  the  shock 
will  be  felt,  since  animal  substances  are  better  conductors  than 
wood.     It  is  necessary  to  remark,  however,  that  when  the  charge 
is  very  intense  or  the  quantity  great,  as  in  the  battery,  then  some 
portion  of  the  fluid  will  escape  from  the  discharging  wire  and 
pass  through  the  hand.     In  such  cases,  therefore,  it  is  prudent  to 
make  use  of  the  discharging  rod. 

Lightning,  in  striking  a  building,  usually  takes  a  course  which 
indicates  the  preference  of  the  fluid  for  the  best  conductors. 

623.  The  electric  fluid  will  sometimes  take  a  shorter  route  through 
a  worse  conductor,  in  preference  to  a  longer  route  through  a  better 
conductor.     The  sparks  will  pass  through  a  short  space  of  air, 
instead  of  following  a  small  wire  thirty  or  forty  feet.     The  pref- 
erence of  the  shorter  route  is  sometimes  indicated  in  taking  the 
electric  shock.     While  one  person  is  receiving  the  shock  from 
the  Leyden  jar,  another  may  grasp  his  arm  without  feeling  the 
least  effect  from  the  charge. 

624.  The  course  of  the  charge  is  frequently  determined  by  the 
influence  of  points,  either  in  dissipating  or  in  receiving  the  fluid. 
Sharp  points,  connected  with  the  best  conductors,  greatly  favor 
the  dispersion  of  the  fluid  during  its  passage,  and  sharp  pointed 
conductors  determine  the  charge  toward  them,  from  a  great  dis- 
tance around.     The  finest  needle,  held  in  the  hand  toward  the 
knob  of  one  of  the  jars  of  a  charged  battery,  will  silently  dis- 
charge it,  in  a  few  seconds  ;  and  if  we  apply  one  hand  to  the 
outside  of  a  Leyden  jar,  and  with  the  other  bring  a  fine  needle 
slowly  to  the  knob  of  the  jar,  only  a  comparatively  feeble  shock 
will  be  felt,  the  dharge  being  rapidly  dissipated  while  the  needle 
is  approaching  the  knob. 

EFFECTS  OF  ELECTRICITY  UPON  ANIMALS. 

625.  We  have  already  several  times  incidentally  adverted  to 
the  shock  communicated  to  the  animal  system,  when  it  is  brought 


424  NATURAL   PHILOSOPHY. 

into  the  electric  circuit,  so  that  the  charge  passes  through  it. 
We  now  propose  to  consider  this  interesting  part  of  our  science 
more  particularly. 

626.  The  electric  shock  is  received,  whenever  the  animal  system 
is  made  a  part  of  the  conducting  communication  between  the  inside 
and  outside  of  a  charged  Leyden  jar.     A  convenient  method  of 
administering  the  shock,  is  to  place  the  charged  jar  on  a  table, 
resting  immediately  on  a  metallic  plate,*  as  a  plate  of  tin,  lead, 
or  copper  ;  then  grasping  a  metallic  rod  in  each  hand,  touch  one 
of  them  to  the  plate  and  the  other  to  the  knob  of  the  jar,  and  a 
sudden  convulsion  of  the  limbs  or  the  breast  will  be  experienced, 
more  or  less  violent  according  to  the  strength  of  the  charge. 
The  effect  is  greatly  heightened  by  feelings  of  dread  or  appre- 
hension, and  it  may  be  resisted  to  a  considerable  degree  by  a 
voluntary  effort.     A  slight  charge  affects  only  the  fingers  or  the 
wrists ;  a  stronger  charge  convulses  the  large  muscles  above  the 
arm-pits  ;  a  still  greater  charge  passes  through  the  breast,  and 
becomes  in  some  degree  painful.     Electricians,  however,  have 
frequently  ventured  upon  charges  sufficiently  powerful  to  con- 
vulse the  whole  frame. 

627.  The  shock  may  be  communicated  to  any  number  of  persons 
at  once.    <This  is  usually  effected  by  their  joining  hands,  while 
the  first  in  the  series  holds  one  of  the  metallic  rods,  (Art.  653,) 
with  which  he  touches  the  plate  or  outside  of  the  jar,  and  the 
last  in  the  series  holds  the  other  rod,  with  which  he  touches  the 
knob  of  the  jar,  at  which  instant  the  whole  number  receive  the 
shock  at  the  same  moment,  and  that  however  extensive  the 
circle  of  persons  may  be.     The  Abbe  Nollet,  a  celebrated  French 
electrician,  gave  the  shock,  at  once,  to  one  hundred  and  eighty 
of  the  king's  guards,  and  to  all  the  members  of  a  convent,  who 
formed  a  large  community.     All  gave  a  spring  at  the  same  mo- 
ment.    The  strength  of  the  shock,  however,  is  somewhat  dimin- 
ished by  passing  through  a  long  circuit,  some  portion  of  the  fluid 
being  dissipated  on  the  way.f     The  connection,  instead  of  being 
made  by  taking  hold  of  hands,  may  be  formed  between   any 
number  of  persons  A,  B,  C,  D,  &c.  as  follows  :  A  may  touch  his 
foot  to  the  foot  of  B  ;  B  may  take  the  hand  of  C,  who  may 
touch  the  foot  of  D  ;  then  each  of  the  company  will  feel  the 
shock  in  one  arm  and  one  leg,  showing  that  the  fluid  pursues 
the  most  direct  course,  agreeably  to  Art.  650. 

628.  If  the  discharge  from  two  square  feet  of  coated  surface 
be  made  to  pass  through,  the  region  of  the  diaphragm,  a  sudden 

*  It  is  safer  to  employ  such  a  plate  than  to  bring  the  conducting  rod  immediately 
into  contact  with  the  outside  coating  of  the  jar  ;  for  in  such  case,  persons  unaccus- 
tomed to  receive  the  shock,  are  apt  to  overturn  the  jar  and  break  it. 

t  Priestley,  p.  97. 


ELECTRICITY.  425 

convulsive  action  of  the  lungs  produces  a  loud  shout.  A  smaller 
charge  produces  a  violent  fit  of  laughter,  even  in  the  gravest  per- 
sons. A  very  strong  charge,  passed  through  the  diaphragm,  pro- 
duces involuntary  sighing  and  tears,  and  sometimes  brings  on  a 
fainting  fit.*  The  charge  of  a  large  battery  is  sufficient  to  de- 
stroy human  life,  especially  if  it  be  received  through  the  head. 
By  standing  on  the  Insulating  Stool,  which  is  a  stool  with  glass 
feet,  a  person  becomes  an  insulated  conductor,  and  may  be  elec- 
trified like  any  other  insulated  conductor.  A  communication  be- 
ing made  with  the  machine,  the  fluid  pervades  the  system,  but 
excites  hardly  any  sensation  except  a  prickling  of  the  hair, 
which  at  the  same  time  rises  and  stands  erect ;  for  the  hairs,  be- 
ing similarly  electrified,  mutually  repel  each  other. 

While  in  this  situation,  the  human  system  exhibits  the  same 
phenomena  as  the  prime  conductor  when  charged  ;  that  is,  it  at- 
tracts light  bodies,  gives  a  spark  to  conductors  brought  near  it, 
and  communicates  a  slight  shock  to  another  person  who  receives 
the  spark  from  it.  Indeed,  the  same  shock  is  felt  by  both  par- 
ties. 

By  means  of  the  insulating  stool,  the  most  delicate  shocks 
may  be  given ;  for  the  charge  may  be  drawn  oiF  from  any  part, 
by  imperfect  conductors.  Thus,  a  pointed  piece  of  wood  will 
draw  off  the  charge  from  the  eye,  in  a  manner  so  gentle,  as  to  se- 
cure that  tender  organ  against  any  possibility  of  injury.  By  a 
variety  of  conductors,  of  different  powers,  and  by  points  and 
balls,  the  sensations  may  be  accommodated,  with  much  delicacy, 
to  the  state  of  the  patient,  or  to  the  nature  of  the  affected  part. 

629.  The  shock  may  be  communicated 
directly  to  any  individual  part  of  the  sys- 
tem, without  affecting  the  other  parts,  by 
making  that  part  form  a  portion  of  the 
electric  circuit,  between  the  inside  and 
outside  of  a  Leyden  jar.  Thus,  let  it  be  re- 
quired to  electrify  an  arm.  Two  directors 
(consisting  of  wires  terminating  in  brass 
knobs,  and  insulated  by  glass  handles)  are 
connected  by  chains  with  the  knob,  and 
the  outside  coating  of  a  charged  jar  ;  then, 
on  applying  one  of  the  directors  to  the 
hand,  and  the  other  to  the  naked  shoulder,  the  arm  is  convulsed. 
In  cases  where  the  patient  requires  only  a  moderate  shock,  the 
charge  is  regulated  by  a  contrivance  attached  to  the  jar,  called 
Lane's  Discharging  Electrometer,  represented  in  Fig.  223.  S,  is 
a  stick  of  solid  glass ;  B,  R,  two  brass  knobs,  connected  by  a 
wire,  which  slides  back  and  forth  in  such  a  way,  that  it  may  be 

*  Encyc.  Metrop. — Morgan's  Lectures. — Singer's  Elements. 
54 


426  NATURAL    PHILOSOPHY. 

set  at  any  required  distance  from  the  knob  of  the  jar.  Suppose 
that  the  outside  of  the  jar  is  grasped  with  the  left  hand,  and  the 
knob  R  with  the  right,  while  the  knob  A  is  constantly  receiving 
sparks  from  the  prime  conductor.*  When  the  knob  B  is  in  close 
contact  with  A,  no  shock  will  be  received,  because  the  two  sides 
of  the  jar  are  in  conducting  communication.  But  when  B  is  re- 
moved a  little  way  from  A,  then  the  shock  will  be  felt  when  the 
charge  has  become  sufficiently  intense  for  the  spark  to  pass  from 
A  to  B  ;  and  the  shock  will  be  greater  or  less  according  as  the 
knobs  are  nearer  to  each  other,  or  further  apart,  until  the  dis- 
tance becomes  too  great  for  the  jar  to  discharge  itself.  When 
the  sliding  wire  is  withdrawn  a  small  distance  from  the  knob  of 
the  jar,  a  constant  succession  of  slight  shocks  are  experienced, 
which  are  sometimes  called  vibrations. 

630.  It  has  already  been  mentioned,  that  life  may  be  destroyed 
by  strong  electrical   charges.     Experiments   have   been  made 
with  the  view  of  investigating  the  nature  of  this  destructive  ac- 
tion.    Dr.  Van  Marum,  of  Haarlem,  selected  for  this  purpose 
eels,  which,  as  is  well  known,  retain  signs  of  irritability,  when 
cut  into  three,  four,  or  six  parts,  and  even  when  deprived  of  their 
heads.     The  eels  employed  in  these  experiments,  were  a  foot 
and  a  half  in  length,  and  the  shock  was  conveyed  through  the 
whole  body.     They  were  instantly  killed,  and  never  moved  af- 
terward.    They  were  immediately  skinned,  and  trial  was  made 
by  pinching,  pricking,  &c.,  but  no  traces  of  irritability  remained. 
When  the  shock  was  made  to  pass  through  individual  parts,  for 
example  the  head,  these  alone  lost  their  irritability,  while  the 
rest  retained  it.     When  the  head  was  kept  free  from  the  shock, 
the  remaining  parts  only  were  paralyzed,  f 

It  had  been  remarked,  that  whenever  animals  had  been  killed 
by  lightning,  the  process  of  spontaneous  putrefaction  ensued  with 
unusual  rapidity.  This  subject  was  examined  by  M.  Achard  of 
Berlin,  by  numerous  experiments.  From  these  it  appeared  that 
electricity  accelerates  putrefaction,  since  it  was  found  that  ani- 
mals recently  killed,  and  animal  substances,  such  as  raw  beef, 
became  putrid  much  sooner  when  electrified.  General  credit  is 
given  to  the  foregoing  experiments,  but  it  seems  easy  to  account 
for  the  increased  tendency  of  milk  to  sour,  and  of  meat  to  be- 
come putrid,  during  a  thunder  storm,  from  the  effects  of  heat 
and  moisture,  which  are  known  and  adequate  causes  of  these 
phenomena. 

631.  Soon  after  the  discovery  of  the  Leyden  jar,  commenced 

*  It  will  be  convenient  to  attach  one  chain  to  the  knob  at  R,  and  another  to  the 
outside  of  the  jar,  near  the  bottom,  and  to  hold  both  in  the  two  hands,  standing  on 
the  insulated  stool. 

t  Nicholson's  Jour.  VIII,  319.— Encyc.  Metrop.,  Art.  Electricity. 


ELECTRICJTY.  427 

the  application  of  electricity  to  medicine ;  and  Medical  Electri- 
city became  thenceforth  a  distinct  branch  of  the  science.  The 
first  cure  said  to  have  been  effected  by  this  agent,  was  upon  a 
paralytic.  Electricity  shortly  became  very  celebrated  for  the 
cure  of  this  disorder,  and  patients  nocked  in  great  numbers  to 
the  practitioners  of  this  branch  of  the  profession.  As  usual,  the 
effects  of  this  new  remedy  were  greatly  exaggerated,  and  it  was 
widely  extolled,  not  only  for  the  cure  of  palsy,  but  of  all  other 
diseases.*  It  was  even  pretended,  that  the  virtues  of  the  most 
valuable  medicines  might  be  conveyed  into  the  system  through 
the  medium  of  electricity,  preserving  their  specific  properties  in 
the  same  manner  as  when  taken  by  the  way  of  the  stomach. 
Preparations  of  this  kind  were  called  Medicated  Tubes.  Pavati, 
an  Italian,  and  Winkler,  a  German,  were  especially  celebrated 
for  this  species  of  practice.  The  mode  was,  to  enclose  the  medi- 
cines in  a  glass  tube,  then  to  excite  the  tube,  and  with  it  to  elec- 
trify the  patient.  In  this  way,  it  was  said,  the  healing  virtues  of 
the  medicine  were  communicated  to  the  system  in  a  manner  at 
once  efficacious  and  agreeable.f 

Pretensions  so  extravagant  could  not  long  be  sustained,  and 
the  natural  consequence  was,  that  the  use  of  electricity  in  medi- 
cine,soon  fell  into  great  neglect,  and  has  remained  in  this  situa- 
tion to  the  present  time.  There  are,  however,  certain  properties 
inherent  in  this  agent,  which  deserve  the  attention  of  the  en- 
lightened physician,  and  inspire  the  hope  that,  in  judicious  hands, 
it  may  still  be  auxiliary  to  the  healing  art.  First,  the  great  ac- 
tivity of  the  agent,  particularly  the  facility  and  energy  with 
which  it  can  be  made  to  act  upon  the  nervous  system,  indicate 
that  it  has  naturally  important  relations  to  medicine.  The 
power  of  being  applied,  locally,  to  any  part  of  the  system,  ren- 
ders it  a  convenient  application  in  cases  where  other  local  rem- 
edies cannot  be  administered.  Secondly,  the  acknowledged 
property  of  electricity  to  promote  the  circulation  of  fluids 
through  capillary  tubes,  Art.  581,  (7,)  suggests  the  probability  of 
its  being  efficacious  in  promoting  the  circulation  of  the  fluids  of 
the  animal  system,  and  in  increasing  the  quantity  of  insensible 
perspiration.  Thirdly,  in  the  history  of  medical  electricity  are 
recorded  well-attested  cures,  effected  by  means  of  electricity,  of 
such  diseases  as  palsy,  rheumatism,  gout,  indolent  tumors,  deaf- 
ness, and  a  variety  of  other  disorders.^ 

*  Priestley,  p.  409.  t  Ibid.,  146. 

t  See  Priestley's  History  of  Electricity,  pp.  146  and  408 — Singer's  Elements,  p. 
292— Phil.  Transactions,  passim— Encyc.  Metropolitana,  Elec.  105— Cavallo,  Com- 
plete Treatise,  Vol.  II. 


NATURAL  PHILOSOPHY. 


CHAPTER  VI. 

OF  THE  CAUSE  OF  ELECTRICAL  PHENOMENA. 

632.  FOR  the  sake  of  convenience,  and  for  the  purpose  of 
avoiding  repetition  and  circumlocution,  we  have  made  frequent 
use  of  the  phrase  electric  fluid.     It  may  be  proper  now  to  inquire, 
whether  there  are  any  just  grounds  for  supposing  such  a  fluid  or 
fluids  to  be  present  in  electrical  phenomena. 

There  are  two  modes  by  which  the  existence  of  such  a  fluid 
may  be  rendered  probable  :  the  first  is,  by  showing  that  such  a 
supposition  is  conformable  to  the  analogy  of  nature  ;  the  second 
is,  by  proving  that  the  agent  of  electrical  phenomena  exhibits 
the  properties  of  a  fluid. 

633.  First,  there  are  some  reasons  derived  from  analogy  for  be- 
lieving in  the  existence  of  an  electric  fluid.     (1.)  The  reasons  in 
favor  of  supposing  that  light  and  heat  are  caused  by  the  agency 
of  peculiar  fluids,  (arguments,  however,  that  we  cannot  discuss 
here,)  which  have  induced  a  general  belief,  are  for  the  most  part 
equally  applicable  to  electricity.     (2.)  In  the  present  state  of  our 
knowledge,  the  most  subtile  of  all  fluids,  indeed  the  most  atten- 
uated form  of  matter,  is  hydrogen  gas,  of  which  one  hundred  cu- 
bic inches  weigh  only  two  and  a  quarter  grains,  being  nearly 
fourteen  times  lighter  than  common  air.    But  at  no  distant  period, 
means  had  not  been  devised  by  mankind  for  proving  the  mate- 
riality of  common  air,  nor  even  of  identifying  the  existence  of 
the  other  gases,  which  now  bear  so  conspicuous  a  part  in  experi- 
mental philosophy.  But  as  knowledge  and  experimental  research- 
es have  advanced,  a  series  of  fluids  still  more  subtle  than  air  have 
come  to  light,  until  we  have  reached  a  body  nearly  fourteen  times 
lighter  than  air,  at  which,  at  present,  the  series  stops.     Is  it  prob- 
able, however,  that  nature  stops  in  her  processes  of  attenuation 
precisely  at  the  point  where,  for  want  of  more  delicate  instru 
ments,  or  more  refined  and  powerful  organs  of  sensation,  our 
methods  of  investigation  and  powers  of  discrimination  come  to 
their  limit  ?     An  examination  of  the  general  analogies  of  nature 
will  lead  us  to  think  otherwise.     The  subordination  which  exists 
among  the  different  classes  of  bodies  that  compose  the  other  de- 
partments of  nature,  is  endless,  or  at  least,  indefinite.     In  the  ani- 
mal creation,  for  example,  beginning  with  the  mammoth  or  the 
elephant,  we  descend  through  numerous  tribes  to  the  insect  which 
is  barely  visible  in  the  sunbeam.     Before  human  ingenuity  had 
devised  means  of  aiding  the  powers  of  vision,  the  naturalist  might 


ELECTRICITY.  429 

have  fixed  this  as  the  limit  of  the  animal  creation.  But  the  in- 
vention of  the  microscope  has  carried  the  range  of  human  vision 
immeasurably  further  ;  and  at  each  successive  improvement  in 
that  instrument,  new  tribes  of  insects  or  animalcules  have  been 
revealed  to  the  eye,  still  more  and  more  attenuated.  A  similar 
subordination  might  be  found  in  the  vegetable  kingdom,  and  in 
the  organic  structure  of  both  animals  and  vegetables. 

634.  To  apply  this  analogy  to  the  case  before  us,  we  begin  the 
series  of  inorganic  bodies  with  platinum,  and  descend  through 
classes  of  bodies  constantly  diminishing  in  density,  until  we  come 
to  ether,  the  lightest  of  liquids,  and  on  the  confines  of  those 
bodies  which  are  invisible  to  the  eye,  and  manifested  only  by 
the  effects  which  they  produce.     By  modern  discoveries  the  se- 
ries has  been  extended  to  hydrogen,  a  body  two  hundred  and 
forty-seven  thousand  times  lighter  than  platinum.     Here  for  the 
present  we  pause,  standing  in  the  same  relation  with  respect  to 
any  fluids  that  may  lie  beyond,  that  the  ancients  stood  with  re- 
spect to  common  air,  and  all  the  other  aeriform  fluids. 

Considerations  of  this  nature  lead  us  to  believe  that  there  are 
in  nature  fluids  more  subtle  than  hydrogen  ;  and  such  being  the 
fact,  we  can  hardly  resist  the  belief,  that  heat,  light,  and  elec- 
tricity, are  bodies  of  this  class, — bodies  which  make  themselves 
known  to  us  by  the  most  palpable  and  energetic  effects,  although 
their  own  constitution  is  too  subtle  and  refined  for  our  organs  to 
recognise,  or  our  instruments  to  identify  them  as  material. 

635.  Secondly,  in  addition  to  the  foregoing  presumptions  in 
favor  of  the  supposition  that  electricity  is  a  peculiar  fluid,  it  ex- 
hibits in  itself  the  properties  of  a  fluid.     The  rapidity  of  its  mo- 
tions, the  power  of  being  accumulated,  as  in  the  Leyden  jar, 
its  unequal  distribution  over  the  surfaces  of  bodies,  (Art.  589,) 
its  power  of  being  confined  to  the  surfaces  of  bodies  by  the  pres- 
sure of  the  atmosphere,  its  attractions  and  repulsions,  are  seve- 
rally properties  which  we  can  hardly  ascribe  to  any  thing  else 
than  an  elastic  fluid  of  the  greatest  tenuity. 

But  granting  the  presence  of  an  elastic  fluid  in  electrical  phe- 
nomena, it  remains  to  be  determined  whether,  according  to  the 
hypothesis  of  Franklin,  these  phenomena  are  to  be  ascribed  to 
the  agency  of  a  single  fluid,  or  whether,  according  to  that  of  Du 
Fay,  they  imply  the  existence  of  two  distinct  fluids.  The  nu- 
merous facts  with  which  the  learner  has  been  made  acquainted 
in  the  preceding  pages,  will  fit  him  to  appreciate  the  evidence 
offered  in  favor  of,  or  against  these  hypotheses  respectively. 

636.  The  principles  of  each  hypothesis  have  been  already  ex- 
plained, (see  Art.  569,)  and  they  have  been  rendered  familiar  by 
repeated  application     It  will  be  recollected  that  they  concur  in 


430  NATURAL   PHILOSOPHY. 

supposing  that  all  bodies  are  endued  with  a  certain  portion  of 
electricity,  called  their  natural  share,  in  which  the  fluid,  whether 
single  or  compound,  is  in  a  state  of  perfect  equilibrium ;  and 
that  in  the  process  of  excitation,  this  equilibrium  is  destroyed. 
But  here  the  two  views  begin  to  diverge  :  the  one  supposes  that 
this  equilibrium  is  destroyed  in  consequence  of  the  separation  of 
two  fluids,  which,  like  an  acid  and  an  alkali  combining  to  form 
a  neutral  salt,  exactly  neutralize  each  other  by  mutual  satura- 
tion, but  which,  when  separated,  exhibit  their  individual  proper- 
ties ;  the  other,  that  the  equilibrium  is  destroyed,  like  that  of  a 
portion  of  atmospheric  air,  by  greater  or  less  exhaustion  on  the 
one  side,  or  condensation  on  the  other.  In  the  former  case,  more- 
over, the  equilibrium  is  restored  by  the  reunion  of  the  two  con- 
stituent fluids  ;  in  the  latter,  by  the  movement  of  the  redundant 
portion  to  supply  the  deficient,  as  air  rushes  into  the  exhausted 
receiver  of  an  air-pump. 

It  is  a  remarkable  fact,  that  nearly  every  electrical  phenome- 
non may  be  perfectly  explained  in  accordance  with  either  hy- 
pothesis ;  nor  is  it  agreed,  that  an  experimentum  crucis*  has  yet 
been  found,  f 

637.  One  of  the  latest  advocates  of  the  hypothesis  of  a  single 
fluid  is  Mr.  Singer,J  an  able  practical  electrician,  and  the  most 
distinguished  defender  of  the  doctrine  of  two  fluids  is  M.  Biot.§ 
In  support  of  the  former  doctrine,  are  offered  such  arguments  as 
the  following.  (1.)  Its  greater  simplicity.  It  is  supposed  to  be 
more  conformable  to  the  Newtonian  rule  of  philosophizing,  "  to 
assign  no  more  causes  than  are  just  sufficient  to  account  for  the 
phenomena."  The  known  frugality  of  nature  in  all  her  opera- 
tions, might  lead  us  to  suppose  that  she  would  not  employ  two 
agents  to  effect  a  given  purpose,  when  a  single  agent  would  be 
competent  to  its  production.  This  argument,  however,  cannot 
be  applied,  either  where  one  cause  is  not  sufficient  to  account 
for  the  phenomena,  or  where  there  is  direct  proof  of  the  exist- 
ence of  more  agents  than  one.  (2.)  The  appearance  of  a  cur- 
rent, circulating  from  the  positive  to  the  negative  surface,  anal- 
ogous to  the  passage  of  air  of  greater  density  into  a  rarefied 
space.  This  point  is  much  insisted  on  by  Singer,  and  numerous 
examples  are  brought  forward  where  the  progress  of  such  a  cur- 
rent is  manifest  to  the  senses.  Thus,  the  flame  of  a  candle, 
brought  into  the  circuit  between  the  inside  and  outside  of  a  Ley- 

*  The  "  experimentum  crucis,"  is  a  phrase  introduced  by  Lord  Bacon,  implying  a 
fact  .which  can  be  explained  on  one  of  two  opposite  hypotheses,  and  not  on  the  other. 
The  figure  is  derived  from  a  cross  set  up  where  two  roads  meet,  to  tell  the  traveller 
which  road  to  take. 

t  Lib.  Useful  Knowledge. 

t  Elements  of  Electricity  and  Electro-Chemistry,  by  George  John  Singer,  London, 
1814. 

§  Traite"  de  Physique,  tome  II. 


ELECTRICITY.  431 

den  jar,  is,  on  the  discharge  of  the  jar,  bent  toward  the  nega- 
tive side  ;  a  pith  ball,  under  similar  circumstances,  moves  in  the 
same  direction  when  a  charged  jar  is  placed  under  the  receiver 
of  an  air-pump,  and  the  air  is  exhausted,  a  luminous  cloud  flows 
from  the  positive  to  the  negative  side,  in  whichever  way  the  jar 
is  electrified.  None  of  these  arguments,  however,  are  found  to 
be  conclusive  ;  for  the  mechanical  effects,  which  are  here  as- 
cribed to  an  elastic  fluid,  that  is,  the  electric  fluid,  flowing  to- 
ward the  negative  side,  can  all  be  accounted  for,  either  upon  the 
principles  of  attraction  and  repulsion,  common  to  both  hypoth- 
eses, or  from  the  mechanical  impulse  of  a  current  of  air,  which  is 
known  to  be  repelled  from  a  point  positively  electrified.  The 
electric  spark  passing  instantaneously,  or  at  least  with  a  veloci- 
ty entirely  inappreciable,  it  is  impossible  to  determine  its  direc- 
tion. 

638.  The  fact  that  bodies  negatively  electrified  repel  each  other, 
(Art.  570,)  is  a  strong  argument  against  the  truth  of  the  hypothesis 
under  consideration.     It  is  not  difficult  to  conceive  that  a  self- 
repellent  fluid  should  communicate  the  same  property  to  two 
pith  balls  in  which  it  resided  ;  but  that  the  mere  deficiency  of  the 
fluid  should  produce  the  same  effect  is  incredible.    This  fact  drove 
^Epinus  (a  celebrated  German  electrician,  who  brought  this  hypo 
thesis  to  the  test  of  mathematical  demonstration)  to  the  necessity 
of  supposing  that  unelectrified  matter  is  self-repellent — a  supposi- 
tion which  is  not  only  destitute  of  proof,  but  which  is  inconsist- 
ent with  the  general  laws  of  nature,  from  which  it  appears  that 
attraction  and  not  repulsion  exists  mutually  between  all  kinds  of 
bodies.     In  the  distribution  of  electricity  upon  surfaces  differing 
in  shape  and  dimensions,  the  fluid  is  found  to  arrange  itself  in 
strict  accordance  with  hydrostatic  principles,  and  that  too  in  bo- 
dies negatively  as  well  as  positively  electrified.     Now  that  the 
privation,  or  mere  absence  of  a  fluid,  should  exhibit  such  prop- 
erties of  a  present  fluid,  is  inconceivable. 

639.  In  favor  of  the  doctrine  of  two  fluids,  the  following  ar- 
guments are  urged.     (1.)  Two  opposite  currents  are  supposed  to 
be  sometimes  indicated.     Thus,  (Art.  619,)  a  card  perforated  by 
a  strong  electric  discharge,  exhibits  burs  or  protrusions  on  both 
sides.     The  appearance  of  the  electric  spark  passing  between 
two  knobs,  is  supposed  by  some  writers  to  indicate  the  meeting 
of  two  fluids  from  opposite  parts.     When  the  spark  is  short,  the 
whole  distance  between  the  two  knobs  through  which  it  passes, 
is  illuminated.    But  when  the  spark  is  long,  those  portions  of  it 
which  are  nearest  to  the  knobs,  are  much  brighter  than  the  cen- 
tral portions.     Near  the  knobs  the  color  is  white,  but  toward 
the  center  of  the  spark  it  is  purplish.     Indeed,  if  the  spark  is 
very  long,  the  middle  part  of  it  is  not  illuminated  at  all,  or  only 


432  NATURAL   PHILOSOPHY. 

very  slightly.  Now  this  imperfectly  illuminated  part,  is  obvious 
ly  the  spot  where  the  two  electricities  unite,  and  it  is  in  conse- 
quence of  this  union,  that  the  light  is  so  imperfect.*  (2.)  The 
two  electricities  are  characterized  by  specific  differences.  The 
light  afforded  by  the  vitreous  surface  is  different  from  that  of  the 
resinous  ;  when  the  two  opposite  portions  of  the  spark  meet,  as 
above,  the  place  of  meeting  is  only  half  the  distance  from  the 
negative  that  it  is  from  the  positive  side  ;  the  bur  protruded  from 
the  card  is  larger  in  the  direction  of  the  vitreous  than  in  that  ot 
the  resinous  fluid ;  and  the  two  severally  produce  certain  chemi- 
cal effects  in  bodies  which  are  peculiar  to  each.  (3.)  But  the 
most  conclusive  argument  in  favor  of  two  fluids,  is  the  perfect 
manner  in  which  this  supposition  accounts  for  the  distribution  of 
electricity  on  bodies  of  different  dimensions.  (See  Arts.  587 — • 
590.)  On  the  hypothesis,  that  electrical  phenomena  are  owing 
to  the  agencies  of  two  fluids,  both  perfectly  incompressible,  the  par- 
ticles of  which  possess  perfect  mobility,  and  mutually  repel  each  oth- 
er, while  they  attract  those  of  the  opposite  fluid,  with  forces  varying 
in  the  inverse  ratio  of  the  square  of  the  distance, — on  this  hypothe- 
sis, M.  Poisson,  a  celebrated  mathematician  of  France,  applied 
the  exhaustless  resources  of  the  calculus,  to  determine  the  vari- 
ous conditions  which  electricity  would  assume  in  distributing 
itself  over  spheres,  spheroids,  and  bodies  of  various  figures.  The 
results  at  \vhich  he  arrived  were  such  as  accord  in  a  very  re- 
markable degree  with  experiment,  and  leave  little  doubt  that  the 
hypothesis  on  which  they  were  built  must  be  true.  Nor  is  any 
supposition  involved  in  the  hypothesis  itself  inconsistent  with 
established  facts.  (4.)  Finally,  authority  is,  at  the  present  day, 
almost  wholly  on  the  side  of  the  doctrine  of  two  fluids — an  opin- 
ion which  has  constantly  gained  new  adherents  with  every  new 
discovery  in  the  science  of  electricity,  particularly  in  the  depart- 
ment of  galvanism. 


CHAPTER  VII. 

OF  ATMOSPHERICAL  ELECTRICITY— THUNDER  STORMS- 
LIGHTNING  RODS. 

640.  HAVING  learned  the  laws  of  electricity  from  a  great  va- 
riety of  experiments,  the  student  is  now  prepared  to  look  upon 
the  works  of  Nature,  and  to  study  the  phenomena  which  the 
same  agent  produces  there  on  a  most  extensive  scale. 

*  Thomson. 


ELECTRICITY.  433 

The  atmosphere  is  always  more  or  less;  electrified.  This  fact  is 
ascertained  by  several  different  forms  of  apparatus.  For  the 
lower  regions,  it  is  sufficient  to  elevate  a  metallic  rod  a  few  feet 
in  length,  pointed  at  the  top,  and  insulated  at  the  bottom.  With 
the  lower  extremity  is  connected  an  electrometer,  which  indi- 
cates the  presence  and  intensity  of  the  electricity.  For  experi- 
ments on  the  electricity  of  the  upper  regions,  a  kite  is  employed, 
not  unlike  a  boy's  kite,  with  the  string  of  which  is  intertwined 
a  fine  metallic  wire.  The  lower  end  of  the  string  is  insulated 
by  fastening  it  to  a  support  of  glass,  or  by  a  cord  of  silk.  But 
as  experiments  of  this  kind  involve  some  personal  hazard,  we 
subjoin,  from  an  excellent  treatise  on  practical  electricity,*  a  few 
directions  for  the  construction  of  this  apparatus. 

641.  An  electric  kite  should  be  constructed  in  the  most  simple 
manner,  for  it  is  an  apparatus  very  liable  to  be  injured  or  lost ; 
its  size  should  be  moderate,  as  there  is  not  often  sufficient  wind 
to  raise  one  that  is  very  large,  which  is  besides  on  several  other 
accounts  very  troublesome  to  manage.     An  ordinary  paper  kite, 
about  four  feet  in  height,  and  two  feet  wide,  varnished  with  dry- 
ing oil  to  defend  it  from  the  rain,  is  sufficiently  well  adapted  to 
this  purpose.     The  string  must  be  made  with  a  thin  copper  or 
silver  thread,  (such  as  is  used  for  gilt  lace,)  interwoven  with  the 
twine  of  which  it  is  formed,  through  its  whole  length.     When 
the  kite  is  raised,  the  string  is  insulated  by  attaching  it  to  a  silk 
cord,  whose  opposite  extremity  may  be  fastened  to  a  rail,  or  any 
fixed  or  heavy  body.     The  end  of  the  metallic  string  is  to  be 
connected  with  an  insulated  conductor,  and  at  two  inches  from 
the  extremity  of  this  conductor,  a  brass  ball,  well  connected  with 
the  ground,  or  the  nearest  water,  is  to  be  placed ;  so  that  when 
the  electricity  becomes  sufficiently  intense  to  pass  an  interval  of 
two  inches,  it  will  be  conducted  safely  away  without  injury  to 
the  experimenter,  who  should  be  cautious,  in  such  cases,  not  to 
approach  the  insulated  conductor ;  but  if  he  has  occasion  to  re- 
move any  apparatus  to  or  from  it,  to  do  so  by  the  aid  of  long  in- 
sulating handles  or  forceps.f 

642.  A  few  facts  may  be  mentioned  to  show  the  hazard  at- 
tending this  class  of  experiments.     Cavallo,  on  one  occasion,  had 
raised  a  kite,  the  string  of  which  was  insulated  by  silk  lace.     A 
cloud  was  over  head,  and  the  electricity  began  to  be  abundant, 
with  which  he  charged  a  pair  of  Leyden  jars.     In  order  to  pre- 
vent any  accident  which  might  arise  from  too  great  an  accumu- 
lation of  the  fluid,  he  wished  to  take  off  the  insulating  silk,  and 
connect  the  string  immediately  with  the  ground.     For  this  pur- 
pose, he  took  hold  of  the  string  and  detached  it  from  its  support. 

*  Singer's  Elements.  t  Singer,  p.  267. 

55 


434  NATURAL    PHILOSOPHY. 

"  While  I  effected  this,  (says  he,)  which  took  up  less  than  half  a 
minute  of  time,  I  received  about  a  dozen  or  fifteen  very  strong 
shocks,  which  I  felt  all  along  my  arms,  in  my  breast,  and  legs  ; 
shaking  me  in  such  a  manner,  that  I  had  hardly  power  enough 
to  effect  my  purpose,  and  to  warn  the  people  in  the  room  to  keep 
their  distance."*  Professor  Richman,  of  Petersburgh,  a  distin- 
guished devotee  of  our  science,  fell  a  victim  to  his  temerity.  He 
had  constructed  an  apparatus  for  observations  on  atmospherical 
electricity,  which  was  entirely  insulated,  and  had  no  contrivance 
for  discharging  it  when  electrified  too  strongly.  On  the  6th  of 
August,  1753,  he  was  examining  the  electricity  of  this  appara- 
tus in  company  with  a  friend ;  while  attending  to  an  experiment, 
his  head  accidentally  approached  the  insulated  rod,  when  his  at- 
tendant observed  a  globe  of  blue  fire,  as  he  called  it,  as  big  as 
his  fist,  jump  from  the  rod  to  the  head  of  the  professor,  which, 
at  that  instant,  was  about  a  foot  from  it.  M.  Richman  was  kill- 
ed instantly ;  a  red  spot  was  left  on  his  forehead,  his  shoe  was 
burst  open,  and  part  of  his  waistcoat  singed  ;  his  companion  was 
benumbed,  and  rendered  senseless  for  some  time  ;  and  the  door- 
case of  the  room  was  split,  and  the  door  torn  off  its  hinges. 

643.  The  most  powerful  apparatus  ever  employed  for  atmo- 
spheric electricity,  was  constructed  in  France  by  M.  de  Romas. 
He  procured  a  kite  seven  feet  long  and  three  feet  wide,  and  ele- 
vated it  to  the  height  of  five  hundred  and  fifty  feet.     A  cloud 
coming  over,  the  most  striking  and  powerful  electrical  phenome- 
na presented  themselves.     Light  straws  that  happened  to  be  on 
the  ground  near  the  string  of  the  kite,  began  to  erect  themselves, 
and  to  perform  a  dance  between  the  apparatus  and  the  ground, 
after  the  manner  of  dancing  images,  as  exhibited  in  ordinary  elec- 
trical experiments.     Art.  581,  (5).     At  length  streams  of  fire  be- 
gan to  dart  to  the  ground,  some  of  which  were  an  inch  in  diam- 
eter, and  ten  feet  long,  exhibiting  the  most  terrific  appearance. 

The  foregoing  facts  evince  the  abundance  of  electricity  in  the 
atmosphere  at  particular  periods  ;  but  experiments  of  a  less  for- 
midable kind  have  been  instituted,  to  ascertain  the  electrical 
changes  of  the  air.  For  this  purpose,  Mr.  Canton,  an  English 
philosopher,  constructed  an  ingenious  apparatus,  which  warned 
him  of  the  presence  of  any  unusual  quantity  of  electricity,  by- 
causing  it  to  ring  a  bell  connected  with  the  lower  extremity  of 
the  apparatus. 

644.  Obvious  as  is  the  connection  between  the  phenomena  of 
common  electrical  apparatus,  and  those  exhibited  in  the  heavens 
during  a  thunder  storm,  yet  the  identity  of  lightning  with  the 
electric  spark,  was  not  dreamed  of  by  the  earlier  electricians. 

•  Cavallo's  Complete  Treatise  on  Electricity,  II,  22 


ELECTRICITY.  435 

To  Dr.  Franklin,  is  universally  conceded  the  merit  of  having  es- 
tablished this  fact,  first  by  reasoning  on  just  principles  of  analo- 
gy, and  afterward  by  actually  bringing  down  the  lightning  from 
the  skies.  The  resemblances  between  the  appearances  of  light- 
ning and  electricity,  were  thus  enumerated. 

(1.)  The  zigzag  form  of  lightning  corresponds  exactly  in  ap- 
pearance with  a  powerful  electric  spark,  that  passes  through  a 
considerable  interval  of  air. 

(2.)  Lightning  most  frequently  strikes  such  bodies  as  are  high 
and  prominent,  as  the  summits  of  hills,  the  masts  of  ships,  high 
trees,  towers  and  spires.  So  the  electric  fluid,  when  striking 
from  one  body  to  another,  always  passes  through  the  most  prom- 
inent parts. 

(3.)  Lightning  is  observed  to  strike  most  frequently  into  those 
substances  that  are  good  conductors  of  electricity,  such  as  met- 
als, water,  and  moist  substances ;  and  to  avoid  those  that  are 
non-conductors. 

(4.)  Lightning  inflames  combustible  bodies ;  the  same  is  ef- 
fected by  electricity. 

(5.)  Metals  are  melted  by  a  powerful  charge  of  electricity : 
this  phenomenon  is  one  of  the  most  common  effects  of  a  stroke 
of  lightning. 

(6.)  The  same  may  be  observed  of  the  fracture  of  brittle 
bodies. 

(7.)  Lightning  has  been  known  to  strike  people  blind :  Dr. 
Franklin  found,  that  the  same  effect  is  produced  on  animals,  by  a 
strong  electric  charge. 

(8.)  Lightning  destroys  animal  life ;  Dr.  Franklin  killed  tur- 
keys of  about  ten  pounds  weight,  by  a  powerful  electric  shock. 

(9.)  The  magnetic  needle  is  affected  in  the  same  way  by 
lightning  and  by  electricity,  and  iron  may  be  rendered  magnetic 
by  both  causes.  The  phenomena  therefore  are  strictly  analo- 
gous, and  differ  only  in  degree  ;  but  if  an  electrified  gun-barrel 
will  give  a  spark,  and  produce  a  loud  report  at  two  inches  dis- 
tance, what  effect  may  not  be  expected  from  10,000  acres  of  elec- 
trified cloud  ?  But  (said  Franklin)  to  ascertain  the  accuracy  of 
these  ideas,  let  us  have  recourse  to  experiment.  Pointed  bodies 
receive  and  transmit  electricity  with  facility ;  let  therefore  a 
pointed  metal  rod  be  elevated  into  the  atmosphere  and  insulated ; 
if  lightning  is  caused  by  the  electricity  of  the  clouds,  such  an 
insulated  rod  will  be  electrified  whenever  a  cloud  passes  over  it ; 
this  electricity  may  be  then  compared  with  that  obtained  in  our 
experiments.* 

645.  Such  were  the  suggestions  of  this  admirable  philosopher ; 
they  soon  excited  the  attention  of  the  electricians  of  Europe,  and 

*  Singer. 


436  NATURAL    PHILOSOPHY. 

having  attracted  the  notice  of  the  King  of  France,  the  approba- 
tion he  expressed  excited  in  several  members  of  the  French 
Academy,  a  desire  to  perform  the  experiment  proposed  by  Frank- 
lin, and  several  insulated  metallic  rods  were  erected  for  that  pur- 
pose. On  the  10th  of  May,  1752,  one  of  these,  a  bar  of  iron 
forty  feet  high  situated  in  a  garden  at  Marly,  became  electrified 
during  the  passage  of  a  stormy  cloud  over  it ;  and  during  a  quar- 
ter of  an  hour  it  afforded  sparks,  by  which  jars  were  charged  and 
other  electrical  experiments  performed.  During  the  passage  of 
the  cloud,  a  loud  clap  of  thunder  was  heard,  so  that  the  identity 
of  these  phenomena  was  thus  completely  proved.  Similar  expe- 
riments were  made  by  several  electricians  in  England. 
.  Doctor  Franklin  had  not  heard  of  these  experiments,  and  was 
waiting  the  erection  of  a  spire  at  Philadelphia  to  admit  an  op- 
portunity of  sufficient  elevation  for  his  insulated  rod,  when  it 
occurred  to  him  that  a  kite  would  obtain  more  ready  access  to 
the  regions  of  thunder  than  any  elevated  building.  He  accord 
ingly  adjusted  a  silk  handkerchief  to  two  light  strips  of  cedar, 
placed  crosswise  ;  and  having  thus  formed  a  kite,  with  a  tail  and 
loop,  at  the  approach  of  the  first  storm,  he  repaired  to  a  field,  ac- 
companied by  his  son.  Having  launched  his  kite,  with  a  pointed 
wire  fixed  to  it,  he  waited  its  elevation  to  a  proper  height,  and 
then  fastened  a  key  to  the  end  of  the  hempen  cord,  and  attached 
this  by  means  of  a  silk  lace  (which  served  to  insulate  the  whole 
apparatus)  to  a  post.  The  first  sign  of  electricity  which  he  per- 
ceived, was  the  separation  of  the  loose  fibres  of  the  hempen  cord  : 
a  dense  cloud  passed  over  the  apparatus,  and  some  rain  falling, 
the  string  of  the  kite  became  wet ;  the  electricity  was  then  col- 
lected by  it  more  copiously,  and  a  knuckle  being  presented  to 
the  key,  a  stream  of  acute  and  brilliant  sparks  was  obtained. 
With  these  sparks,  spirits  were  fired,  jars  charged,  and  the  usual 
electrical  experiments  performed.  Thus  was  the  identity  of  light- 
ning and  electricity,  which  had  been  indicated  by  so  many  anal- 
ogies, now  established  by  the  most  decisive  experiment. 

646.  It  is  a  matter  of  much  importance  to  the  science  of  Me- 
teorology, to  ascertain  from  what  source  atmospherical  electricity 
originates.  Among  the  known  sources  of  this  agent  none  seems 
so  probable,  as  the  evaporation  and  condensation  of  watery  vapor. 
We  have  the  authority  of  two  of  the  most  able  and  accurate  phi- 
losophers, Lavoisier  and  La  Place,  for  stating  that  bodies  in  pass- 
ing from  the  solid  or  liquid  state  to  that  of  vapor,  and,  conversely 
in  returning  from  the  aeriform  condition  to  the  liquid  or  solid  state, 
give  unequivocal  signs  of  either  positive  or  negative  electricity.* 

*  Dr.  Thomson,  in  his  Outlines  of  Electricity,  makes  the  following  note. — M.  Pou- 
illet  has  lately  published  a  set  of  experiments,  which  seems  to  overturn  Volta's  the- 
ory of  the  evolution  of  electricity  by  evaporation.  He  has  shown  that  no  electricity 
is  evolved  by  evaporation;  unless  some  chemical  combination  takes  place  at  the  same 


ELECTRICITY. 


437 


Combustion  is  also  attended  with  the  evolution  of  electricity, 
and  even  the  friction  of  opposite  currents  of  wind,  or  of  a  high 
wind  against  opposing  objects,  probably  generates  more  or  less  of 
the  same  agent.  The  production  of  electricity  during  evapora- 
tion and  condensation,  may  be  rendered  evident  by  Coulomb's 
electrical  balance  ;  as  may  that  evolved  during  the  friction  of  air. 
If  the  stem  of  a  tobacco  pipe  be  heated  red  hot,  and  a  drop  of 
water  be  introduced  by  way  of  the  bowl,  the  jet  of  steam  falling 
on  the  brass  ball  (Fig.  216,  a,)  of  the  balance  will  electrify  it,  so 
that  it  will  set  the  index  of  the  balance  in  motion. 

It  is  obvious,  that  a  cause  which  produces  only  very  feeble 
signs  of  electricity  in  so  small  a  quantity  of  vapor  as  that  which 
arises  from  a  single  drop  of  water,  may  still  be  sufficient  to  occa- 
sion a  vast  accumulation  of  the  same  agent,  in  such  a  quantity 
of  vapor  as  that  which  is  daily  ascending  into  the  atmosphere  ; 
for  it  has  been  calculated,  that  more  than  two  thousand  millions 
of  hogsheads  of  water  are  evaporated  from  the  Mediterranean 
alone  in  one  summer's  day.* 

THUNDER    STORMS,  f 

647.  The  following  are  the  leading  facts  respecting  the  elec- 
tricity of  the  atmosphere  in  relation  to  this  subject,  and  these 
are  facts  which  have  been  established  by  numerous  observers, 
of  the  most  accurate  and  diligent  class.  Beccaria,  an  Italian 
electrician,  continued  his  observations  on  the  electricity  of  the 
atmosphere  for  fifteen  years  with  the  greatest  assiduity ;  and 
Cavallo,  Read,  Saussure,  and  others,  prosecuted  the  same  inqui- 
ries with  similar  zeal. 

(1.)  Thunder  clouds  are,  of  all  atmospheric  bodies,  the  most 
highly  charged  with  electricity  ;  but  all  single,  detached,  or  in- 
sulated clouds,  are  electrified  in  greater  or  less  degrees,  some- 
times positively  and  sometimes  negatively.  When,  however,  the 
sky  is  completely  overcast  with  a  uniform  stratum  of  clouds,  the 
electricity  is  much  feebler,  than  in  the  single  detached  masses 
before  mentioned.  And,  since  fogs  are  only  clouds  near  the 
surface  of  the  earth,  they  are  subject  to  the  same  conditions :  a 
driving  fog,  of  limited  extent,  is  often  highly  electrified.J 

time.  But  it  follows  from  his  experiments,  that  electricity  is  evolved  abundantly 
during  combustion,  the  burning  body  giving-  out  resinous,  and  the  oxygen  vitreous 
electricity.  In  like  manner,  the  carbonic  acid  emitted  by  vegetables,  is  charged  with 
resinous  electricity,  and  the  oxygen  (probably)  charged  with  vitreous  electricity. — 
Thomson's  Outlines,  p.  440.  But  we  shall  be  slow  to  reject  the  results  of  experi- 
ments performed  by  such  experimenters  as  Lavoisier  and  La  Place,  especially  when 
confirmed  by  the  testimony  of  Volta  and  Saussure. 

*  Singer. 

t  The  most  complete  enumeration  of  facts  hitherto  made  respecting  thundef 
storms,  was  published  by  M.  Arago  in  the  Annuaire  for  1838. 

I  Ed.  Encyc.,  VIII,  310. 


438  NATURAL   PHILOSOPHY. 

(2.)  The  electricity  of  the  atmosphere  is  strongest  when  hot 
weather  succeeds  a  series  of  rainy  days,  or  when  wet  weather 
succeeds  a  series  of  dry  days  ;  and  during  any  single  day,  the  air 
is  most  electrical  when  the  dew  falls  before  sunset,  or  when  it 
begins  to  exhale  before  sunrise. 

(3.)  In  clear,  steady  weather,  the  electricity  generally  remains 
positive ;  but  in  falling  or  stormy  weather,  it  is  constantly 
changing  from  positive  to  negative,  or  from  negative  to  posi- 
tive.* 

648.  Such  are  the  circumstances  of  atmospheric  electricity  in 
general ;  next,  let  us  attend  to  the  peculiar  phenomena  of  thun- 
der storms,  chiefly  as  they  are  exhibited  in  our  own  climate. 

(1.)  In  thunder  storms  there  is  usually  a  singular  and  powerful 
combination  of  all  the  elements, — of  darkness,  rain,  thunder  and 
lightning,  and  sometimes  hail. 

(2.)  They  occur  chiefly  in  the  hottest  season  of  the  year,  and 
after  mid-day  ;  and  are  more  frequent  and  violent  in  warm  than 
in  cold  countries. 

(3.)  Thunder  storms  never  occur  beyond  75°  of  latitude — sel- 
dom beyond  650.f 

(4.)  In  this  state,  (Connecticut,)  thunder  storms  usually  come 
from  the  west,  either  directly,  or  from  the  northwest  or  south- 
west ;  but  occasionally  from  the  east. 

(5.)  Violent  thunder  and  lightning  are  frequently  seen  in  vol- 
canoes and  water  spouts. 

(6.)  Thunder  storms  sometimes  descend  almost  to  the  surface 
of  the  sea,  and  fall  upon  the  sides  of  mountains  ;  in  which  case, 
they  are  extremely  violent. 

(7.)  We  occasionally  .observe  the  following  circumstances 
succeed  each  other  in  regular  order :  first,  a  vivid  flash  of  light- 
ning,— then  a  loud  peal  of  thunder, — and,  after  a  short  interval, 
a  sudden  fall  of  rain,  which  sometimes  stops  as  suddenly  as  it 
began.J 

649.  There  are   in   thunder  storms,  evidently,  two  distinct 
classes  of  phenomena  to  be  accounted  for.     The  first  class  con- 
sists of  the  common  elements  of  a  storm, — clouds,  wind,  and 
rain ;   the  second,  of  thunder  and  lightning.      The  following 
proposition  embraces,  in  our  view,  the  true  explanation  of  both 
these  classes  of  phenomena  : 

The  storm  itself,  including  every  thing  except  the  electrical  ap- 
pearances, is  produced  in  the  same  manner  as  other  storms  of  wind 

*  Singer,  p.  273.  t  Arago,  Annuaire,  1838,  p.  389. 

t  Morgan's  Lectures  on  Electricity  contain  an  excellent  view  of  the  natural  agen- 
cies of  electricity. 


ELECTRICITY.  439 

and  rain  ;  and  the  electricity,  and  of  course  the  thunder  and  light- 
ning, is  owing  to  the  rapid  condensation  of  watery  vapor.* 

We  do  not,  therefore,  consider  electricity  as  the  cause,  but  as 
the  consequence  of  the  storm  ;  or  as  a  concomitant  of  the  clouds, 
wind,  and  rain. 

A  sudden  and  copious  deposition  of  condensed  vapor,  is  an  es- 
sential preliminary,  or  concomitant,  of  a  thunder  storm,  since 
when  the  process  of  condensation  is  slow,  too  much  of  the  elec- 
tricity evolved  would  escape  to  allow  of  the  requisite  accumula- 
tion ;  and  the  amount  of  vapor  condensed  must  be  copious,  else 
the  quantity  of  the  electric  fluid  produced  would  not  be  sufficient 
to  cause  the  violent  phenomena  of  a  thunder  storm.  These  con- 
ditions imply,  first,  that  the  air  is  extremely  humid,  or  the  dew- 
point  (Art.  483 )  very  high,  so  that  a  slight  reduction  of  tem- 
perature will  precipitate  vapor  ;  and,  secondly,  that  the  air  which 
affords  the  vapor,  or  materials  of  the  storm,  is  suddenly  cooled. 
The  cooling  may  be  conceived  to  take  place  in  several  different 
ways, — as  the  meeting  of  hot  and  cold  bodies  of  air  by  opposite 
winds,  or  the  sudden  transference  of  hot  and  humid  air  from  the 
surface  of  the  earth  to  the  region  of  congelation  by  the  agency 
of  tornadoes  or  whirlwinds.  All  that  we  require  is  that  the  re- 
duction of  temperature  should  be  great  and  sudden.f 

650.  The  earth  itself,  in  its  natural  state,  is  a  vast  conductor, 
where  any  excess  of  the  electric  fluid  may  readily  discharge 
itself.  Accordingly,  where  a  cloud  highly  charged  comes  near 
to  the  earth,  it  puts  the  latter  in  the  opposite  electrical  state  by 
induction,  and  a  discharge  takes  place  between  the  earth  and 
the  cloud.  When  the  electricity  which  is  expelled  from  the 
earth  by  the  approach  of  a  cloud,  returns  to  it,  it  sometimes  pro- 
duces a  violent  shock,  known  by  the  name  of  the  returning 
stroke^  (Art.  575.)  Indeed,  in  some  instances,  lightning  is  sup- 
posed to  take  a  circuitous  route  in  its  way  from  one  cloud  to  an- 
other, first  darting  to  the  earth  and  thence  to  the  opposite  cloud, 
the  distance  of  the  clouds  from  each  other  being  too  great  to 
permit  the  discharge  through  the  intervening  space  of  air.  And 
since  electricity  passes  quietly  without  light  or  noise,  when  it 
makes  its  way  through  good  conductors,  and  manifests  its  splen- 
dors and  mechanical  energies  only  when  its  path  is  obstructed 
by  imperfect  conductors,  it  is  reasonably  inferred  that  the  light- 

*  Other  causes,  such  as  friction,  change  of  temperature,  &c.,  may  have  some  in- 
fluence, but  the  condensation  of  vapor,  producing  electricity,  which  is  accumulated  in 
insulated  clouds,  (thunder  clouds  being  insulated  by  the  circumambient  air,)  is  to  be 
regarded  as  the  chief  source  of  the  electricity  of  thunder  storms. 

t  The  great  amount  of  electricity  which  is  found  to  be  evolved  from  the  cloud  of 
steam  suddenly  condensed  from  the  pipe  of  a  locomotive,  favors  the  explanation  here 
given  of  the  origin  of  the  electricity  of  thunder  storms.  (See  Experiments  of  Dr. 
Patterson  and  others,  Franklin  Journal,  Feb.  1842.) 

i  Mahon's  (Earl  Stanhope's)  "  Principles  of  Electricity.'- 


440  NATURAL   PHILOSOPHY. 

ning  and  thunder  have  an  origin  extrinsic  to  the. fluid  itself;  that 
the  lightning  is  produced  by  the  sudden  and  powerful  condensa- 
tion which  the  air  experiences  when  compressed  before  the  fluid, 
(a  known  cause  of  heat  and  light,)  and  that  the  thunder  is  pro- 
duced by  the  collapsing  of  the  air,  filling  the  sudden  void,  occa- 
sioned by  the  passage  of  the  fluid,  (a  known  cause  of  sound.)* 
The  zigzag  appearance  of  lightning  is  well  explained  by  sup- 
posing the  air  so  much  condensed  before  it,  as  to  turn  its  course 
in  another  direction,  where  the  same  resistance  is  again  experi- 
enced and  another  change  encountered.  This  explanation  is  ren- 
dered the  more  probable  by  experiments,  which  show  that  the  zig- 
zag appearance  is  very  much  increased  when  the  electric  spark 
is  passed  through  condensed  air,  but  disappears  entirely  when  it 
is  passed  through  a  vacuum.  (Arts.  610,  613.) 

651.  If  we  now  apply  these  principles  to  the  facts  before  enu- 
merated, (Art.  647,)  we  shall  find  them  capable  of  a  clear  and 
satisfactory  explanation. 

All  insulated  clouds  are  electrical  in  a  greater  or  less  degree, 
because  their  very  formation  implies  a  condensation  of  watery 
vapor,  and  the  state  of  insulation  prevents  the  escape  of  the 
electric  fluid,  that  is  thus  evolved.  The  electricity  is  stronger 
in  such  insulated  detached  clouds,  some  of  which  are  positive, 
and  some  negative,  than  in  a  sky  uniformly  overcast,  because  in 
the  latter  case  the  opposite  electricities  are  neutralized,  while  in 
the  former  they  are  kept  separate.  The  electricity  of  the  at- 
mosphere is  strongest  when  hot  weather  succeeds  a  series  of 
rainy  days,  and  when  wet  weather  succeeds  a  series  of  dry  days, 
because  then,  in  both  cases,  the  evaporation  is  most  sudden  and 
abundant ;  and  on  a  single  day,  the  signs  of  electricity  are  strong- 
est at  the  rising  and  falling  of  the  dew,  that  being  the  very  mo- 
ment when  the  evaporation  in  the  morning,  and  the  condensa- 
tion in  the  evening,  are  most  copious.  Thunder  showers  are 
most  frequent  and  violent  in  hot  climates,  and  during  the  hottest 
seasons  of  the  year,  for  in  such  places  and  at  such  times,  the 
causes  supposed  are  in  most  active  operation.!  Electricity,  if 
evolved  at  all  by  slower  processes  of  evaporation  and  condensa- 
tion, finds  its  equilibrium  before  it  can  accumulate  in  sufficient 
quantity  to  produce  the  phenomena  of  a  thunder  storm.  Thunder 
storms  usually  occur  after  mid-day,  because  it  is  chiefly  during 
the  hottest  part  of  the  day,  or  a  little  after  it,  that  the  meeting 
of  those  opposite  currents  occurs,  which  generate  the  storm ; 
since  it  is  at  the  places  of  greatest  rarefaction  that  this  concourse 
of  winds  takes  place  ;  and  therefore  during  the  heat  of  summer, 

*  Cavallo,  Complete  Treatise,  p.  274. 

t  See  a  tabular  view  of  the  thunder  storms  of  different  climates,  Arago,  Annuaire, 
1838,  p.  403. 


ELECTRICITY.  441 

the  sun  is  sometimes  followed  round  the  globe  by  a  succession 
of  thunder  storms. 

In  volcanoes,  the  most  vivid  lightnings  and  the  heaviest  thun- 
ders are  produced,  because  here  an  immense  quantity  of  heated 
vapor  is  thrown  out,  which,  on  reaching  the  cold  regions  of  the 
atmosphere,  is  suddenly  condensed  into  thick  clouds ;  and  the 
same  phenomena  are  often  terrific  in  water  spouts,  because  here 
the  sudden  formation  of  clouds  and  rain,  occasions  a  vast  evolu- 
tion of  electricity.  Violent  thunder  storms  sometimes  fall  upon 
the  sides  of  mountains,  or  upon  the  surface  of  the  sea,  for  here, 
on  account  of  the  proximity  of  the  clouds,  the  discharges  are 
made  toward  the  earth  or  sea,  which,  in  ordinary  cases,  are 
made  from  cloud  to  cloud. 

652.  All  the  foregoing  facts  appear  to  admit  of  a  clear  ex- 
planation, in  conformity  with  the  supposition,  that  the  storm 
itself,  including  all  the  phenomena  except  the  electrical,  is  pro- 
duced like  other  storms  of  wind  and  rain,  by  the  sudden  cooling 
of  heated  air,  charged  with  watery  vapor,  (Art.  487,)  and  that 
the  electrical  phenomena  are  produced  by  the  condensation  of  the 
vapor  itself  into  clouds  and  rain.  But  the  last  fact  mentioned 
appears  to  present  greater  difficulties.  We  refer  to  that  quick 
succession  of  events,  Art.  648,  (7,)  occurring  in  the  following  or- 
der ;  namely,  first,  a  vivid  flash  of  lightning — then  a  loud  peal 
of  thunder — and,  after  a  little  interval,  a  sudden  fall  of  rain, 
which  frequently  stops  as  suddenly  a!s  it  commenced.  At  first 
view,  it  would  seem  that  the  rain  which  follows  the  electrical 
discharge  is  produced  by  it ;  whereas,  according  to  the  foregoing 
views,  the  lightning  is  not  the  cause,  but  rather  a  consequence 
of  the  formation  of  the  rain.  (Art.  649.)  But  suppose  that  the 
events  were  to  take  place  as  required  by  our  principles ;  that 
drops  of  rain  were  suddenly  to  coalesce,  forming  a  shower,  and 
that  the  attendant  lightning  and  thunder  were  produced  by  this 
process  ;  let  us  see  in  what  order  the  notice  of  these  events  would 
reach  the  earth.  The  passage  of  light  being  nearly  instantane- 
ous, the  flash  would  be  seen  the  instant  of  the  explosion  ;  but 
sound  is  a  comparatively  slow  traveller,  and  would  take  its  own 
time  to  reach  the  ear  ;  and  rain,  a  slower  traveller  still,  would 
arrive  much  later  than  the  other  two.  To  submit  these  succes- 
sive events  to  something  like  mathematical  calculation,  we  will 
suppose  the  cloud  to  be  one  fourth  of  a  mile  high,  and  that  the 
precipitation  of  the  rain,  and  the  evolution  of  the  electricity, 
which  causes  the  explosion,  are  cotemporaneous  events.  First, 
the  fash  would  reach  us  without  any  perceptible  interval.  Sec- 
ondly, the  sound  travelling  at  the  rate  of  1130  feet  per  second, 
would  require  1.15  seconds  to  reach  the  ear.  Thirdly,  the  rain, 
descending  like  any  other  falling  body,  we  may  calculate  its 
time  accordingly.  The  times  being  as  the  square  roots  of  the 

56 


442  NATURAL    PHILOSOPHY. 

spaces,  \/16.1  :  1  :  : \/l'32Q  :  9  seconds.  The  time  would  be  con- 
siderably more  than  this,  on  account  of  the  resistance  of  the 
air.  Our  principles,  therefore,  require  that  the  flash,  the  report, 
and  the  shower,  should  succeed  each  other  in  the  order  in  which 
they  actually  occur. 

LIGHTNING    RODS. 

653.  Dr.  Franklin  had  no  sooner  satisfied  himself  of  the  iden- 
tity of  electricity  and  lightning,  than,  with  his  usual  sagacity,  he 
conceived  the  idea  of  applying  the  knowledge  acquired  of  the 
properties  of  the  electric  fluid,  so  as  to  provide  against  the  dan- 
gers of  thunder  storms.     The  conducting  powers  of  metals,  and 
the  influence  of  pointed  bodies,  to  collect  and  transmit  the  fluid, 
naturally  suggested  the  structure  of  the  Lightning  Rod.     The 
experiment  was  tried,  and  has  proved  completely  successful ;  and 
probably  no  single  application  of  scientific  knowledge  ever  se- 
cured more  celebrity  to  its  author. 

654.  Lightning  rods  are  at  present  usually  constructed  of 
wrought  iron,  about  three  fourths  of  an  inch  in  diameter.     The 
parts  may  be  made  separate,  but,  when  the  rod  is  in  its  place, 
they  should  be  joined  together  so  as  to  fit  closely,  and  to  make  a 
continuous  surface,  since  the  fluid   experiences  much  resistance 
in  passing  through  links  and  other  interrupted  joints.     At  the 
bottom,  the  rod  should  terminate  in  two  or  three  branches,  going 
off'  in  a  direction  from  the  building.     The  depth  to  which  it 
enters  the  earth  should  not  be  less  than  five  feet ;  but  the  neces- 
sary depth  will  depend  somewhat  on  the  nature  of  the  soil :  wet 
soils  require  a  less,  and  dry  soils  a  greater  depth.     In  dry  sand 
it  must  not  be  less  than  ten  feet ;  and  in  such  situations,  it  would 
be  better  still  to  connect,  by  a  convenient  conducting  communi- 
cation, the  lower  end  of  the  rod  with  a  well  or  spring  of  water. 
It  is  useful  to  fill  up  the  space  around  the  part  of  the  rod  that 
enters  the  ground,  with  coarsely  powdered  charcoal,  which  at 
once  furnishes  a  good  conductor,  and  preserves  the  metal  from 
corrosion.     The  rod  should  ascend  above  the  ridge  of  the  building 
to  a  height  determined  by  the  following  principle :  that  it  will 
protect  a  space  in  every  direction  from  it,  whose  radius  is  equal  to 
twice  its  height.     It  is  best,  when  practicable,  to  attach  it  to  the 
chimney,  which  needs  peculiar  protection,  both  on  account  of 
its  prominence,  and  because  the  products  of  the  combustion, 
smoke,  watery  vapor,  &c.  are  conductors  of  electricity.     For  a 
similar  reason  a  kitchen  chimney,  being  that  in  which  the  fire 
is  kept  during  the  season  of  thunder  storms,  requires  to  be 
especially  protected.     The  rod   is  terminated  above   in  three 
forks,  each  of  which  ends  in  a  sharp  point.     As  these  points  are 
liable  to  have  their  conducting  power  impaired  by  rust,  they  are 


ELECTRICITY.  443 

protected  from  corrosion  by  being  covered  with  gold  leaf;  or 
they  may  be  made  of  solid  silver  or  platina.  Black  paint,  being 
made  of  charcoal,  forms  a  better  coating  for  the  rod  than  paints 
made  of  other  colors,  the  bases  of  which  are  worse  conductors. 
The  rod  may  be  attached  to  the  building  by  wooden  stays.  Iron 
stays  are  sometimes  employed,  and  in  most  cases  they  would  be 
safe,  since  electricity  pursues  the  most  direct  route,  (Art.  623  ;) 
but  in  case  of  an  extraordinary  charge,  there  is  danger  that 
it  will  divide  itself,  a  part  passing  into  the  building  through  the 
bolt,  especially  if  this  terminates  in  a  point.  Buildings  furnished 
with  lightning  rods  have  occasionally  been  struck  with  lightning ; 
but  on  examination  it  has  generally,  if  not  always,  been  found 
that  the  structure  of  the  rod  was  defective  ;  or  that  too  much 
space  was  allotted  for  it  to  protect.  When  the  foregoing  rules 
are  observed,  the  most  entire  confidence  may  be  reposed  in  this 
method  of  securing  safety  in  thunder  storms. 

. 
I 


CHAPTER  VIII. 


PRECAUTIONS   FOR   SAFETY   DURING   THUNDER   STORMS.— ANL 
MAL  ELECTRICITY.— CONCLUDING  REMARKS. 

655.  THE  great  number  of  pointed  objects  that  rise  above  the 
general  level,  in  a  large  city,  have  the  effect  to  dissipate  the  elec- 
tricity of  a  thunder  cloud,  and  to  prevent  its  charge  from  being 
concentrated  on  any  single  object*  Hence,  damage  done  by 
lightning  is  less  frequent  in  a  populous  town,  than  in  solitary 
buildings.  For  similar  reasons,  a  great  number  of  ships,  lying  at 
the  docks,  disarm  the  lightning  of  its  power,  and  thus  avert  the 
injury  to  which  the  form  of  their  masts  would  otherwise  expose 
them.  A  solitary  ship  on  the  ocean,  unprotected  by  conductors, 
would  appear  to  be  peculiarly  in  danger  from  lightning  ;  but. 
while  the  greater  number  of  ships  that  traverse  the  ocean  are 
wholly  unprotected,  accidents  of  this  kind  are  comparatively  rare. 
The  reason  probably  is,  that  water  being  a  better  conductor  than 
wood,  the  course  of  the  discharge  toward  the  water  is  not  easily 
•diverted,  and  will  not  take  the  mast  in  its  way  unless  the  latter 
lies  almost  directly  in  its  course.  Barns  are  peculiarly  liable  to 
be  struck  with  lightning,  and  to  be  set  on  fire  ;  and  as  this  occurs 
at  a  season  when  they  are  usually  filled  with  hay  and  grain,  the 
damage  is  more  serious,  for  the  quantity  of  combustible  matter 
they  contain,  is  such  as  to  render  the  fire  unmanageable.  Pro- 
fessor Silliman  ascribes  this  liability  of  barns  to  be  struck  with 
lightning,  to  the  influence  of  the  evaporation  that  proceeds  from 


444  NATURAL   PHILOSOPHY. 

the  fresh  hay,  &c.,  which  is  supposed  to  furnish  a  conducting 
medium  like  the  smoke  of  a  chimney.* 

656.  Silk  dresses  are  sometimes  worn  with  the  view  of  pro- 
tection, by  means  of  the  insulation  they  afford.     They  cannot, 
however,  be  deemed  very  effectual  unless  they  completely  envel- 
op the  person  :  for  if  the  head  and  the  extremities  of  the  limbs 
are  exposed,  they  will  furnish  so  many  avenues  to  the  fluid  as  to 
render  the  insulation  of  the  other  parts  of  the  system  of  little 
avail.     The  same  remark  applies  to  the  supposed  security  that 
is  obtained  by  sleeping  on  a  feather  bed.     Were  the  person 
situated  within  the  bed,  so  as  to  be  entirely  enveloped  by  the 
feathers,  they  would  afford  some  protection ;  but  if  the  person  be 
extended  on  the  surface  of  the  bed,  in  the  usual  posture,  with  the 
head  and  feet  nearly  in   contact  with  the  bedsted,  he  would 
rather  lose  than  gain  by  the  non-conducting  properties  of  the  bed  ; 
since  being  a  better  conductor  than  the  bed,  the  charge  would 
pass  through  him  in  preference  to  that.f     The  horizontal  posture, 
however,  is  safer  than  the  erect ;  and  if  any  advantage  on  the 
whole  is  gained  by  lying  in  bed  during  a  thunder  storm,  it  proba- 
bly arises  from  this  source.     The  same  principle  suggests  a  reason 
why  men  or  animals  are  so  frequently  struck  with  lightning  when 
they  take  shelter  under  a  tree  during  a  thunder  storm.     The 
fluid  first  strikes  the  tree,  in  consequence  of  its  being  an  elevated 
and  pointed  object,  but  it  deserts  the  tree  on  reaching  the  level 
of  the  man  or  animal,  because  the  latter  is  a  better  conductor 
than  the  tree. 

Tall  trees  situated  near  a  dwelling  house,  furnish  a  partial  pro- 
tection to  the  building,  being  both  better  conductors  than  the 
materials  of  the  house,  and  having  the  advantage  of  superior  ele- 
vation. 

657.  The  protection  of  chimneys  is  of  particular  importance, 
for  to  these  a  discharge  is  frequently  determined.     When  a  fire 
is  burning  in  the  chimney,  the  vapor,  smoke  and  hot  air,  which 
ascend  from  it,  (as  has  been  intimated  in  article  654,)  furnish  a 
conducting  medium  for  the  fluid ;  but  even  when  no  fire  is  burn- 
ing, the  soot  that  lines  the  interior  of  a  chimney  is  a  good  con- 
ductor, and  facilitates  the  passage  of  the  discharge. 

It  is  quite  essential,  during  a  thunder  storm,  to  avoid  every 
considerable  mass  of  water,  and  even  the  streamlets  that  have 
resulted  from  a  recent  shower  ;  for  these  are  all  excellent  con- 

*  American  Journal  of  Science,  Vol.  Ill,  p.  345. 

t  Security  to  the  person  might  be  obtained  by  an  entire  covering  of  either  a  very 
bad  or  a  very  good  conductor.  In  the  former  case,  the  electricity  would  not  approach 
the  system  ;  in  the  latter  case  it  would  confine  itself  to  the  covering.  Clothes  when 
very  wet  have  been  supposed  to  furnish  a  protection  on  this  principle.  (See  an  inter- 
esting case  stated  by  Professor  Hitchcock,  in  the  American  Journal  of  Science.) 


ELECTRICITY.  445 

\ 

ductors,  and  the  height  of  a  human  being,  when  connected  with 
them,  is  very  likely  to  determine  the  course  of  an  electric  dis- 
charge. The  partial  conductors,  through  which  the  lightning 
directs  its  course,  Avhen  it  enters  a  building,  are  usually  the  ap- 
pendages of  the  walls  and  partitions  ;  the  most  secure  situation 
is  therefore  the  middle  of  the  room,  and  this  situation  may  be 
rendered  still  more  secure  by  standing  on  a  glass  legged  stool,  a 
hair  mattress,  or  even  a  thick  woolen  rug.  The  part  of  every 
building  least  liable  to  receive  injury  is  the  middle  story,  as  the 
lightning  does  not  always  pass  from  the  clouds  to  the  earth,  but 
is  occasionally  discharged  from  the  earth  to  the  clouds.  Hence 
it  is  absurd  to  take  refuge  in  a  cellar,  or  in  the  lowest  story  of  a 
house  ;  and  many  instances  are  on  record  in  which  the  basement 
story  has  been  the  only  part  of  the  building  that  has  sustained 
severe  injury.  Whatever  situation  is  chosen,  any  approach  to 
the  fire-place  should  be  particularly  avoided.*  An  open  door  or 
window  is  an  unsafe  situation,  because  the  lightning  is  apt  to 
traverse  the  large  timbers  that  compose  the  frame  of  the  house, 
and  would  be  determined  towards  the  animal  system  on  account 
of  its  being  a  better  conductor.  In  a  carriage  the  passenger  is 
safer  in  the  central  part  than  next  to  the  walls ;  but  a  carriage 
may  be  effectually  protected  by  attaching  to  its  upper  surface 
metallic  strips  connected  with  the  wheel  tire.  The  fillets  of 
silver  plating  which  are  frequently  bound  round  the  carriage, 
may  be  brought  into  the  conducting  circuit. 

ANIMAL   ELECTRICITY. 

658.  Of  the  natural  agencies  of  electricity,  one  of  the  most  re- 
markable, is  that  exhibited  by  certain  species  of  fish,  especially 
the  Torpedo  and  the  Gymnotus.  This  peculiar  property  of  the 
Torpedo  was  known  to  the  ancient  naturalists,  and  is  accurately 
described  by  Aristotle,  and  by  Pliny.  Aristotle  says  that  this 
fish  causes  or  produces  a  torpor  upon  those  fishes  it  is  about  to 
seize,  and  having  by  that  means  got  them  into  its  mouth  it  feeds 
upon  them.  Pliny  says  that  this  fish  if  touched  by  a  rod  or 
spear,  even  at  a  distance,  paralyzes  the  strongest  muscles. 

The  fact,  however,  that  this  extraordinary  power  depends 
upon  electricity,  was  not  known,  until  about  the  year  1773, 
when  it  was  ascertained  by  Mr.  Walsh,  that  the  Torpedo  is 
capable  of  giving  shocks  to  the  animal  system,  analogous  to  those 
of  the  Leyden  jar.  Though  this  property  is  regarded  as  estab- 
lishing the  identity  of  the  power  with  the  electric  fluid,  yet  this 
power,  as  developed  in  the  Torpedo,  has  never  been  made  to  af- 
ford a  spark,  nor  to  produce  the  least  effect  upon  the  most  deli- 

*  Singer. 


446  NATURAL   PHILOSOPHY. 

x 

cate  electrometer.*  As  late  as  the  year  1828,  experiments  were 
made  upon  the  Torpedo,  by  Sir  Humphry  Davy,  and  the  con- 
clusions at  which  he  arrived,  were  that  the  electricity  resides  in 
this  animal  in  a  form  suited  exclusively  to  the  purpose  of  com- 
municating shocks  to  the  animal  system,  while  it  has  little  or 
nothing  else  in  common  with  the  properties  of  electricity,  as  de- 
veloped in  various  artificial  arrangements.! 

659.  The  Torpedo  is  a  flat  fish,  seldom  twenty  inches  in  length, 
but  one  found  on  the  British  coast  was  four  and  a  half  feet  long 
The  electricity  of  the  Torpedo  has  the  same  relation  as  commo; 
electricity  to  bodies  in  respect  to  their  conducting  power,  being 
readily  transmitted  through  metals,  water,  and  other  conductors, 
and  not  being  transmitted  through  glass,  and  other  non-conduct- 
ors. 

The  electric  organs  of  the  Torpedo  are  two  in  number,  and 
placed  one  on  each  side  of  the  cranium  and  gills.  The  length 
of  each  organ  is  somewhat  less  than  one  third  part  of  the  length 
of  the  whole  animal.  Each  organ  consists  of  perpendicular 
columns  reaching  from  the  under  to  the  upper  surface  of  the 
body,  and  varying  in  length  according  to  the  various  thickness  of 
the  flesh  in  different  parts.  The  number  of  these  columns  is  not 
constant,  varying  not  only  in  different  Torpedoes,  but  likewise 
in  different  ages  of  the  animal,  new  ones  seeming  to  be  produced 
as  the  animal  grows.  In  a  very  large  Torpedo,  one  electric 
organ  has  been  found  to  consist  of  one  thousand  one  hundred 
and  eighty-two  columns.  The  diameter  of  a  column  is  about 
one  fifth  of  an  inch.  Each  column  is  divided  by  horizontal  par- 
titions, consisting  of  transparent  membranes,  placed  over  each 
other  at  very  small  distances,  and  forming  numerous  interstices, 
which  appear  to  contain  a  fluid.  The  number  of  partitions  con- 
tained in  a  column  one  inch  in  length,  has  been  found  in  some 
instances  not  less  than  one  hundred  and  fifty.  By  this  arrange- 
ment the  amount  of  electrified  surface  is  exceedingly  great ; 
equivalent  in  one  instance  to  one  thousand  and  sixty-four  feet  of 
coated  glass.  Hence,  the  effects  of  the  electricity  of  the  Tor- 
pedo are  such  as  correspond  to  those,  which,  in  artificial  arrange- 
ments, are  produced  by  diffusing  a  given  quantity  of  fluid  over  a 
great  surface,  by  which  its  intensity  is  much  diminished.J 


*  Humboldt. 

t  Phil.  Trans.  1829.  A  reflection  naturally  suggested  by  this  fact  is,  that  the 
fluid  which  is  excited  in  the  various  species  of  electrical  apparatus,  both  the  common 
and  Voltaic,  is  a  compound,  embracing  several  distinct  substances. 

t  In  the  Philosophical  Transactions  for  1832,  Dr.  Davy  has  given  an  interesting 
series  of  observations  on  the  Torpedo.  He  ascertained  that  the  animal  has  the  power 
of  giving  magnetic  polarity  to  iron,  and  of  affecting  chemical  decompositions  in  a 
slight  degree.  The  shocks  were  very  powerful.  By  giving  repeated  shocks,  the  oldei 
fish  were  rendered  languid  and  died  soon. 


ELECTRICITY.  447 

660.  The  Gymnotus,  or  Surinam  eel,  is  found  in  the  rivers  of 
South  America.     Its  ordinary  length  is  from  three  to  four  feet ; 
but  it  is  said  to  be  sometimes  twenty  feet  long,  and  to  give 
a  shock  that  is  instantly  fatal.    The  electrical  organs  of  the  Gym- 
notus, constitute  more  than  one  third  part  of  the  whole  animal ; 
they  consist  of  two  pairs,  of  different  sizes  and  placed  on  differ- 
ent sides.     The  shock  communicated  to  fishes  instantly  paraly- 
zes them,  so  that  they  become  the  prey  of  the  Gymnotus.     By 
irritating  the  animal  with  one  hand,  while  the  other  is  held  at 
some  distance  in  the  water,  a  shock  is  received  as  severe  as  that 
of  the  Leyden  jar. 

Unlike  the  Torpedo,  the  Gymnotus  gives  a  small  but  percepti- 
ble spark,  affording  additional  proof  of  the  identity  of  the  power 
with  that  of  electricity. 

M.  Humboldt.  in  his  travels  in  South  America,  describes  a  sin- 
gular method  of  catching  the  Gymnotus,  by  driving  wild  horses 
into  a  lake  which  abounds  with  them.  The  fish  are  wearied  or 
exhausted  by  their  efforts  against  the  horses,  and  then  taken  ;  but 
such  is  the  violence  of  the  charge  which  they  give,  that  some 
of  the  horses  are  drowned  before  they  can  recover  from  the  par- 
alyzing shocks  of  the  eels. 

The  Silurus  electricus,  is  a  fish  found  in  some  of  the  rivers  of 
Africa.  Its  electrical  powers  are  inferior  to  those  of  the  Torpedo 
and  Gymnotus,  but  they  are  still  sufficient  to  give  a  distinct 
shock  to  the  human  system. 

661.  Certain  furred  animals,  particularly  the  cat,  become  spon- 
taneously electrified.     This  is  more  especially  observable  on  cold 
windy  nights,  when  the  state  of  the  air  is  favorable  to  insulation. 
At  such  times  a  cat's  back  will  frequently  afford  electrical  sparks. 
Ancient  historians  mention  a  number  of  very  remarkable  occur- 
rences, of  good  or  evil  omen,  which  are  due  to  the  electricity  of 
the  atmosphere.     Herodotus  informs  us  that  the  Thracians  dis- 
armed the  sky  of  its  thunder  by  throwing  their  arms  into  the 
air ;  and  that  the  Hyperboreans  produced  the  same  effect  by 
launching  arrong  the  clouds  darts  armed  with  points  of  iron. 
CaBsar,  in  his  Commentaries,  says  that  in  the  African  war,  after 
a  tremendous  storm  which  threw  the  whole  of  the  Roman  army 
into  great  disorder,  the  points  of  the  darts  of  a  great  number  of 
the  soldiers  shone  with  a  spontaneous  light.     In  the  month  of 
February  (says  he)  about  the  second  watch  of  the  night,  there 
suddenly  arose  a  great  cloud,  followed  by  a  dreadful  storm  of 
hail,  and  in  the  same  night  the  points  of  the  darts  of  the  fifth  le- 
gion appeared  on  fire.* 

During  a  dry  snow  storm,  when  electricity  is  evolved  in  great 
quantities,  and,  on  account  of  the  dry  state  of  the  air,  is  partly 


*  Ed.  Encyc.  VIII,  311.    Arago,  Annuaire,  1838,  p.  375. 


448  NATURAL   PHILOSOPHY. 

insulated  on  conducting  bodies,  similar  appearances  are  exhibited. 
Thus,  the  ears  of  horses,  and  various  pointed  bodies,  emit  faint 
streams  of  light.  These  phenomena  are  sometimes  exhibited  in 
a  most  striking  manner  in  a  storm  at  sea,  when  the  masts  of  a 
ship,  yard-arms,  and  every  other  pointed  object,  are  tipped  with 
lightning.* 

CONCLUDING    REMARKS. 

662.  From  the  energy  which  electricity  displays  in  our  exper- 
iments, and  much  more  in  thunder  storms,  there  can  be  no  ques 
tion  that  it  holds  an  important  rank  among  the  ultimate  causes  of 
natural  phenomena.  Its  actual  agencies,  however,  are  liable  to 
be  misinterpreted,  and  that  they  have  been  so  in  fact,  is  too  mani- 
fest from  the  history  of  the  science.  After  the  splendid  experi- 
ments with  the  Leyden  jar,  and  more  especially  after  the  identity 
of  electricity  with  lightning  had  been  proved,  electricians  fancied 
that  they  had  discovered  the  clue  which  would  conduct  them 
safely  through  the  labyrinth  of  nature.  Every  thing  not  before 
satisfactorily  accounted  for,  was  now  ascribed  to  electricity. 
They  saw  in  it,  not  only  the  cause  of  thunder  storms,  but  of 
storms  in  general ;  of  rain,  snow,  and  hail ;  of  whirlwinds  and 
water  spouts  ;  of  meteors  and  the  aurora  borealis  ;  and  finally, 
of  tides  and  comets,  and  the  motions  of  the  heavenly  bodies,  f 
Later  electricians  have  found  in  the  same  agent  the  main  spring 
of  animal  and  vegetable  life,  and  the  grand  catholicon  which 
cures  all  diseases.  Recent  attempts  have  been  made  to  establish 
the  very  identity  of  galvanic  electricity  and  the  nervous  influ- 
ence, by  which  the  most  important  functions  of  animal  life  are 
controlled.  J 

Among  the  most  important  of  the  agencies  of  electricity  in 
the  economy  of  nature,  is  that  which,  according  to  the  views  of 
Sir  Humphry  Davy,  it  sustains  in  relation  to  the  chemical  agen- 
cies of  bodies.  Chemical  and  electrical  attractions,  he  supposes, 
are  one  and  the  same  thing,  or  at  least  dependent  on  the  same 
cause,  the  attractions  between  the  elements  of  a  compound  aris- 
ing solely  from  their  being  naturally  in  opposite  electrical  states. 
But  the  discussion  of  this  hypothesis  belongs  more  appropriately 
to  galvanism,  a  branch  of  our  subject  which,  on  account  of  its 
peculiarities,  especially  in  the  mode  of  excitation,  has  been  con- 
stituted a  separate  department  of  science. 

It  is  a  remark  of  Lord  Bacon,  that  things  appear  usually  to 
better  advantage  and  more  important  in  their  relations,  than  in 
their  individualities.  "  The  contrary,  (says  he,)  has  made  many 

*  See  Jones's  Sketches  of  Naval  Life,  I,  199. 

t  Encyc.  Brit.,  Electricity. 

I  Wilson  Philip,  Phil.  Trans.     Tilloch's  Phil  Mag.  XXX,  488. 


ELECTRICITY.  449 

particular  sciences  to  become  barren,  shallow,  and  erroneous, 
while  they  have  not  been  nourished  and  maintained  from  a  com- 
mon fountain."  The  truth  of  this  remark  is  strikingly  exhibit- 
ed in  respect  to  electricity.  If  in  its  original  form  it  is  an  inter- 
esting and  wonderful  agent,  still  more  astonishing  and  important 
are  the  properties  it  has  disclosed  in  its  relations  to  certain  chem- 
ical agents,  to  magnetism,  to  heat,  and  to  light,  giving  rise  to  the 
different  sciences  of  Galvanism,  Electro-Magnetism,  Thermo- 
Electricity,  and  to  the  art  of  gilding  and  copying  by  the  Electro- 
type process.  Much,  we  believe,  remains  to  be  discovered  re- 
specting the  useful  purposes  which  this  mysterious  agent  is  des- 
tined to  perform  for  man  ;  but  enough  has  been  revealed  to  as- 
sure us  that  electricity  is  the  agent  by  which  are  to  be  achieved 
the  most  refined  and  delicate  performances  of  art,  and  by  which 
man  is  to  acquire  his  most  perfect  mastery  over  nature.* 

*  In  the  distribution  of  subjects  in  Yale  College,  Galvanism  and  its  kindred  subjects 
are  assigned  to  the  chemical  department. 

The  most  extensive  and  complete  treatise  hitherto  published  on  the  subject  of  Elec- 
tricity in  all  its  relations,  is  the  work  of  Becquerel,  in  French,  consisting  of  seven 
volumes  8vo.  Pouillet,  in  the  first  volume  of  his  Elemens  de  Physique,  gives  a  fuD 
and  able  view  of  these  subjects. 

57 


450  NATURAL  PHILOSOPHY. 


PART    VII. MAGNETISM. 


663.  MAGNETISM  is  the  science  which  treats  of  the  properties  and 
effects  of  the  magnet.     The  same  term  is  also  used  to  denote  the 
unknown  cause  of  magnetic  phenomena  ;  as  when  we  speak  of 
magnetism  as  excited,  imparted,  and  so  on. 

Magnets  are  bodies,  either  natural  or  artificial,  which  have 
the  property  of  attracting  iron,  and  the  power,  when  freely  sus- 
pended, of  taking  a  direction  toward  the  poles  of  the  earth.  The 
natural  magnet  is  sometimes  called  the  loadstone.*  It  is  an  ox- 
ide of  iron  of  a  peculiar  character,  found  occasionally  in  beds 
of  iron  ore.  Though  commonly  met  with  in  irregular  masses 
only  a  few  inches  in  diameter,  yet  it  is  sometimes  found  of  a 
much  larger  size.  One  recently  brought  from  Moscow  to  Lon- 
don, weighed  one  hundred  and  twenty-five  pounds,  and  support- 
ed more  than  two  hundred  pounds  of  iron.f 

664.  The  attractive  powers  of  the  loadstone  have  been  known 
from  a  high  antiquity,  and  are  mentioned  by  Homer,  Pythagoras, 
and  Aristotle.     But  the  directive  powers  were  not  known  in  Eu- 
rope until  the  twelfth  century,  though  some  writers  have  endea- 
vored to  trace  the  history  of  the  compass  needle  to  a  remoter  pe- 
riod, and  some  have  strenuously  maintained,  that  the  Chinese 
were  in  possession  of  it  many  centuries  before  it  was  known  to 
the  Europeans.J 

Magnetism  is  the  most  recent  of  all  the  physical  sciences,  and 
notwithstanding  the  numerous  discoveries  achieved  in  it  within 
a  few  years,  and  the  remarkable  precision  with  which  its  laws 
have  been  ascertained,  yet  it  is  still  to  be  regarded  as  a  science 
quite  in  its  infancy,  although  it  is  rapidly  progressive. 

665.  If  a  magnet  be  rolled  in  iron  filings,  it  will  attract  them 
to  itself.     This  effect  takes  place  especially  at  two  opposite 
points,  where  a  much  greater  quantity  of  the  filings  will  be  col- 
lected than  in  any  other  parts  of  the  body.     The  two  opposite 
points  in  a  magnet,  where  its  attractive  powers  appear  chiefly 

»  Said  to  be  derived  from  l&dan,  a  Saxon  word  which  signifies  to  guide 

t  Partington's  Manual,  II,  243. 

t  Cavallo  on  Magnetism  ;  Barlow,  Encyc.  Metrop. ;  Klaproth,  Amer  Jour,  xl.,  242. 


MAGNETISM.  451 

to  reside,  are  called  its  poles.     The  straight  line  which  joins  the 
poles,  is  called  the  axis.     (See  Fig.  224.) 

Fig.  225. 
Fig.  224.  s 


If  a  large  sewing  needle,  or  a  small  bar  of  steel  be  rubbed  on 
the  loadstone,  one  extremity  on  one  pole,  and  the  other  extremi- 
ty on  the  other,  the  needle  or  bar  will  itself  become  a  magnet, 
capable  of  exhibiting  all  the  properties  of  a  loadstone.  With- 
out staying  at  present  to  describe  more  minutely  the  process  of 
making  artificial  magnets,  we  will  suppose  ourselves  provided 
with  several  magnetic  needles  and  bars,  and  we  may  proceed 
with  them  to  study  the  leading  facts  of  the  science  of  magnet- 
ism. By  attaching  a  fine  thread  to  the  middle  of  a  needle,  and 
suspending  it  so  as  to  move  freely  in  a  horizontal  plane,  or  by* 
resting  it  on  a  point,  as  is  represented  in  Fig.  225,  we  shall  have 
a  simple  and  convenient  apparatus  for  numerous  experiments. 
The  needle  thus  suspended  will  place  itself  in  a  direction  near- 
ly, though  not  exactly,  north  and  south.  If  the  needle  is  drawn 
out  of  the  position  it  assumes  when  at  rest,  it  will  vibrate  on 
either  side  of  that  position  until  it  finally  settles  in  the  same 
line  as  before,  one  pole  always  returning  toward  the  north,  and 
the  other  toward  the  south.  -Hence  the  two  poles  are  denomi- 
nated respectively  north  and  south  poles.  In  magnets  prepared 
for  experiments,  these  poles  are  marked  either  by  the  letters  N 
and  S,  or  by  a  line  drawn  across  the  magnet  near  one  end,  which 
denotes  that  the  adjacent  pole  is  the  north  pole. 

666.  By  means  of  the  foregoing  apparatus,  we  may  ascertain 
that  the  magnet  has  the  following  general  properties,  viz : 

First,  powers  of  attraction  and  repulsion. 

Secondly,  the  power  of  communicating  magnetism  to  iron  or 
steel  by  induction. 

Thirdly,  polarity,  or  the  power  of  taking  a  direction  toward 
the  poles  of  the  earth. 

Fourthly,  the  power  of  inclining  itself  toward  a  point  below 
the  horizon,  usually  denominated  the  dip  of  the  needle. 

The  further  development  of  these  properties  will  constitute 
the  subjects  of  the  following  chapters. 


452  NATURAL   PHILOSOPHY. 


CHAPTER  I. 

OF   MAGNETIC  ATTRACTION. 

667.  WHEN  either  pole  of  a  magnet  is  brought  near  to  a  piece  of 
iron,  a  mutual  attraction  takes  place  between  them. 

Thus,  when  the  ends  of  a  magnetic  bar  or  needle  are  dipped 
into  a  mass  of  iron  filings,  these  adhere  in  a  cluster  to  either 
pole.  A  bar  of  soft  iron,  or  a  piece  of  iron  wire,  resting  on  a 
cork,  and  floating  on  the  surface  of  water  or  quicksilver,  may  be 
led  in  any  direction  by  bringing  near  to  it  one  of  the  poles  of  a 
magnet.  This  action  is  moreover  reciprocal ;  that  is,  the  iron 
attracts  the  magnet  with  the  same  force  that  the  magnet  at- 
tracts the  iron.  If  the  two  bodies  be  placed  on  separate  corks 
and  floated,  they  will  approach  each  other  with  equal  momenta ; 
or  if  the  iron  be  held  fast,  the  magnet  will  move  toward  it. 

668.  Two  other  metals  besides  iron,  namely,  nickel  and  cobalt, 
are  susceptible  of  magnetic  attraction.     These  metals,  however, 
exist  in  nature  only  in  comparatively  small  quantities,  and  there- 
fore by  magnetic  bodies,  are  usually  intended  such  as  are  ferru- 
ginous.    Even  iron,  in  some  of  its  combinations  with  other  bo- 
dies, loses  its  magnetic  properties  ;  indeed,  only  a  few  of  the  nu- 
merous ores  of  iron  are  attracted  by  the  magnet.     But  soft  me- 
tallic iron  and  some  of  the  ores  of  the  same  metal,  affect  the 
needle  even  when  existing  in  exceedingly  small  quantities,  so 
that  the  magnet  becomes  a  very  delicate  test  of  the  presence  of 
iron.     Compass  needles  are  sometimes  said  to  be  disturbed  by 
the  minute  particles  of  steel  left  in  the  dial  plate  by  the  graver  ;* 
and  the  proportion  of  iron  in  some  minerals  may  be  exactly  es- 
timated by  the  power  they  exert  upon  the  needle.f 

669.  In  the  action  of  magnets  on  each  other,  poles  of  the  same 
name  repel,  while  those  of  different  names  attract  each  other. 

Thus,  the  north  pole  of  one  magnet  will  repel  the  north  pole 
of  the  other,  and  attract  its  south  pole.  The  south  pole  of  one 
will  repel  the  south  pole  of  the  other,  and  attract  its  north  pole. 
These  effects,  it  will  be  perceived,  are  analogous  to  those  produ- 
ced by  the  two  species  of  electricity ;  and  they  equally  imply 
two  species  of  magnetism  or  two  magnetic  fluids,  (as  it  is  con- 
venient to  call  them,)  namely,  the  northern  and  the  southern,  or 
as  they  are  now  denominated,  the  boreal  and  the  austral  fluids. 

*  Eaton,  Am.  Jour.  Science,   xiv,  15.  t  Biot. 


MAGNETISM.  453 

A  very  simple  piece  of  apparatus  will  serve  ^  Fig.  226. 
to  exhibit  the  foregoing  property.  The  accom- 
panying figure  represents  two  sewing-needles 
magnetized  and  suspended  by  fine  threads. 
On  approaching  the  north  pole  of  a  magnetic 
bar  to  the  north  poles  of  the  needles,  they  are 
forcibly  repelled ;  but  on  applying  the  south 
pole  of  a  bar  as  in  figure  226,  the  north  poles 
of  the  needles  are  attracted  toward  it. 

670.  By  bringing  a  magnet  near  to  iron  or  steel,  the  latter  is 
rendered  magnetic  by  induction. 

Thus,  let  the  north  pole  of  a  Fig.  227. 

magnetic  bar  A,  (Fig.  227,)  be 
brought  near  to  one  end  of  an  |B 

unmagnetized  bar  of  soft  iron  B  : 
the  iron  will  immediately  be- 
come itself  a  magnet,  capable  of 
attracting  iron  filings,  having  po- 
larity when  suspended,  and  possessing  the  power  of  communi- 
cating the  same  properties  to  other  pieces  of  iron.  It  is,  howev- 
er, only  while  the  iron  remains  in  the  vicinity  of  the  magnet,  that 
it  is  endued  with  these  properties ;  for  let  the  magnet  be  with- 
drawn, and  it  loses  at  once  all  the  foregoing  powers.  This,  it 
will  be  remarked,  is  asserted  of  soft  iron  ;  for  steel  and  hardened 
iron  are  differently  affected  by  induced  magnetism. 

On  examining  the  kind  of  magnetism  induced  upon  the  two 
ends  of  the  iron  bar  B,  (Fig.  227,)  which  we  may  easily  do  by 
bringing  it  near  to  the  poles  of  the  needle,  (Fig.  225,)  we  shall 
llnd  that  the  nearer  end  has  south,  and  the  remoter  end  north 
polarity.  This  effect  also  is  analogous  to  that  produced  by  elec- 
trical induction.  (See  Arts.  595,  596.)  A  corresponding  effect 
would  have  taken  place,  had  the  south,  instead  of  the  north  pole 
of  the  magnet  been  presented  to  the  bar  of  iron  ;  in  which  case, 
the  nearer  end  would  have  exhibited  the  northern,  and  the  remo- 
ter end  southern  polarity.  Or,  to  express  this  important  propo- 
sition in  general  terms, 

Each  pole  of  a  magnet  induces  the  opposite  kind  of  polarity  in 
that  end  of  the  iron  which  is  nearest  to  it,  and  the  same  kind  in  that 
end  which  is  most  remote. 

• 

671.  It  is  not  essential  to  the  success  of  these  experiments, 
that  the  bars  of  iron  which  receive  magnetism  by  induction, 
should  be  placed  in  a  straight  line  with  the  magnet ;  they  may 
be  at  right  angles  to  it,  or  inclined  at  any  other  angle,  the  only 
essential  condition  being,  that  the  end  of  the  bar  should  be  brought 
near  to  the  pole  of  the  magnet.  Indeed,  the  effect  is  increased, 


454  NATURAL   PHILOSOPHY. 

that  is,  the  magnetism  of  the  iron  bar  is  j-jg  228. 

rendered  stronger,  when  the  bar  is  in- 
clined  toward  the  magnet,  as  in  Fig. 
228,  and  is  the  strongest  of  all  when  it 
is  placed  parallel  to  the  magnet;  for  it 
will  be  seen  that  in  these  two  latter  po- 
sitions, both  poles  of  the  magnet  conspire 
in  their  action  upon  the  iron  bar. 

672.  The  power  of  a  magnet  is  increased  by  the  exertion  of  its 
inductive  power  upon  a  piece  of  iron,  in  its  neighborhood. 

The  end  of  the  piece  of  iron  contiguous  to  the  pole  of  the 
magnet,  is  no  sooner  endued  with  the  opposite  polarity,  than  it 
reacts  upon  the  magnet  and  increases  its  intensity  ;  and  a  series 
of  actions  and  reactions  take  place  between  the  two  bodies,  simi- 
lar to  what  occurs  in  electrical  induction.  (See  Art.  596,  &c.) 
On  this  account  the  powers  of  a  magnet  are  increased  by  action, 
and  impaired  or-even  lost  by  long  disuse.  By  adding,  from  time  to 
time,  small  pieces  of  iron  to  the  weight  taken  up  by  a  magnet, 
its  powers  may  be  augmented  greatly  beyond  their  original  amount. 
Hence,  the  force  of  attraction  of  the  dissimilar  poles  of  two  mag- 
nets, is  greater  than  the  force  of  repulsion  of  the  similar  poles : 
because,  when  the  poles  are  unlike,  each  contributes  to  enhance 
the  power  of  the  other,  but  when  they  are  alike,  the  influence 
which  they  reciprocally  exert,  tends  to  make  them  unlike,  and  of 
course  to  impair  their  repulsive  energies.  „ 

Hence,  also,  a  strong  magnet  has  the  power  of  reversing  the 
poles  of  a  weak  one.  Suppose  the  north  pole  of  the  weaker  body 
to  be  brought  into  contact  with  the  north  pole  of  the  stronger  ; 
the  latter  will  expel  north  polarity,  or  the  boreal  fluid,  and  attract 
the  austral,  a  change  which  in  certain  cases  will  be  permanent. 

673.  If  the  north  pole  of  a  magnetic  bar  be  placed  upon  the 
middle  of  an  iron  bar,  the  two  ends  of  the  latter  will  each  have 
north  polarity,  while  the  part  of  the  bar  immediately  in  contact 
with  the  magnet  receives  south  polarity ;  and  if  the  same  north 
pole  be  placed  on  the  center  of  a  circular  piece  of  iron,  all  parts 
of  the  circumference  will  be  endued  with  north  polarity,  while 
the  plate  will  have  a  south  pole  in  the  center.     By  cutting  the 
plate  into  the  form  of  a  star,  each  extremity  of  the  radii  becomes 
a  weak  north  pole  when  the  north  pole  of  a  magnet  is  placed  in 
the  center  of  the  star.     If  an  iron  bar  is  placed  between  the  dis- 
similar poles  of  two  magnetic  bars,  (all  being  in  one  straight  line,) 
both  of  the  magnets  will  conspire  to  increase  the  intensity  of  each 
pole  of  the  bar,  and  the  magnetism  imparted  to  the  bar  will  be 
considerably  stronger  than  from  either  magnet  alone  ;  but  if  the 
same  bar  be  placed  between  the  two  similar  poles,  the  opposite 
polarity  will  be  imparted  to  each  end,  while  the  same  polarity 


MAGNETISM.  455 

is  given  to  the  center  of  the  bar.  Thus  if  the  bar  be  placed 
between  the  north  poles  of  two  magnets,  each  end  of  the  bar 
will  become  a  south  pole  and  the  center  a  north  pole.  When 
one  end  of  a  magnetic  bar  is  applied  to  the  ends  of  two  or  more 
wires  or  sewing  needles,  the  latter  arrange  themselves  in  radii 
diverging  from  the  magnetic  pole.  This  effect  is  in  consequence 
of  their  remoter  ends  becoming  endued  with  similar  polarity,  and 
repelling  each  other.  A  like  effect  is  observable  among  the  fila- 
ments of  iron  filings,  that  form  a  tuft  on  the  ends  of  a  magnetic 
bar. 

674.  The  foregoing  experiments  are  sufficient  to  show  that 
when  a  piece  of  iron  is  attracted  by  the  magnet,  it  is  first  itself 
converted  into  a  magnet  by  the  inductive  influence  of  the  mag- 
netizing body.     Each  of  the  iron  filings  which  compose  the  tuft 
at  the  pole  of  a  magnetic  bar  or  needle,  is  itself  a  magnet,  and  in 
consequence  of  being  such,  induces  the  same  property  in  the  next 
particle  of  iron,  and  that  in  the  next,  and  so  on  to  the  last.    Hence 
magnetic  attraction  does  not  exist,  strictly  speaking,  between  a 
magnet  and  iron,  but  only  between  the  opposite  poles  of  mag- 
nets ;  for  the  iron  must  first  become  a  magnet  before  it  is  capa- 
ble of  magnetic  influence. 

675.  Soft  iron  readily  acquires  magnetism  and  as  readily  loses 
it;  hardened  steel  acquires  it  more  slowly,  but  retains  it  perma- 
nently. 

In  the  preceding  example,  the  magnetism  acquired  by  a  bar  of 
iron,  by  the  process  of  induction,  is  retained  only  so  long  as  the 
magnetizing  body  acts  upon  it.  Soon  after  the  two  bodies  are 
separated,  the  bar  loses  all  magnetic  properties. 

When  a  bar  of  steel  is  placed  very  near  a  strong  magnet,  the 
action  of  the  magnet  commences  immediately  upon  the  end  of 
the  bar  nearest  to  it,  the  north  pole  for  example  communicating 
south  polarity  to  the  contiguous  extremity  of  the  bar.  According 
to  our  previous  experience,  we  should  expect  to  find  the  remote 
end  of  the  bar  a  north  pole  ;  but  such  is  not  the  immediate  result ; 
a  sensible  time  is  required  before  the  north  polarity  is  fully  im- 
parted to  the  remote  extremity.  Indeed,  if  the  bar  be  a  long  one, 
it  sometimes  happens  that  the  north  polarity  never  reaches  the 
farthest  end,  but  stops  short  of  it  at  some  intermediate  point. 
This  north  pole  is  succeeded  by  a  second  south  pole,  that  by  an- 
other north  pole,  and  thus  several  alternations  between  the  two 
poles  occur  before  reaching  the  end  of  the  bar. 

676.  The  process  of  magnetizing  a  steel  bar  or  needle,  is  ac- 
celerated by  any  cause  which  excites  a  tremulous  or  vibratory 
motion  among  the  particles  of  the  steel.     Striking  on  the  bar  with 
a  hammer  promotes  the  process  in  a  remarkable  degree,  especially 


456  NATURAL   PHILOSOPHY. 

if  it  occasions  a  ringing  sound,  which  indicates  that  the  particles 
are  thrown  into  a  vibratory  motion.  The  passage  of  an  electric 
discharge  through  a  steel  bar  under  the  influence  of  a  magnet, 
produces  permanent  magnetism.  Heat  also  greatly  facilitates 
the  introduction  of  the  magnetic  fluid  into  steel.  The  greatest 
possible  degree  of  magnetism  that  can  be  imparted  to  a  steel  bar, 
is  communicated  by  first  heating  the  steel  to  redness,  and  while 
it  is  under  the  influence  of  a  strong  magnet,  quenching  it  suddenly 
with  cold  water. 

A  magnet,  however,  loses  its  virtues  by  the  same  means  as, 
during  the  process  of  induction,  were  used  to  promote  their  ac- 
quisition. Accordingly,  any  mechanical  concussion  or  rough 
usage,  impairs  or  destroys  the  powers  of  a  magnet.  By  falling 
on  a  hard  floor,  or  by  being  struck  with  a  hammer,  it  is  greatly 
injured.  Heat  produces  a  similar  effect.  A  boiling  heat  weak- 
ens and  a  red  heat  totally  destroys  the  power  of  a  needle.  On 
the  other  hand,  cold  augments  the  powers  of  the  magnet ;  in- 
deed, they  improve  with  every  reduction  of  temperature  hither- 
to applied  to  them.* 

As  iron  and  steel  are  found  of  various  degrees  of  hardness,  so 
the  power  of  acquiring  and  of  losing  magnetism,  is  very  vari- 
ous in  different  ferruginous  bodies.  It  is  in  general  true,  that  this 
power  is  in  proportion  to  the  hardness.  Thus,  the  attraction  of 
soft  malleable  iron  for  the  magnet  being  100,  that  of  hard  cast 
steel  is  only  49,  and  that  of  cast  iron  only  48.f 

677.  If  a  steel  bar  rendered  magnetic  by  induction,  be  divided  into 
any  two  parts,  each  part  will  be  a  complete  magnet,  having  two 
opposite  poles. 

We  here  meet  with  a  remarkable  distinction  between  magnet- 
ic and  electric  induction.  When  a  body  electrified  by  induction, 
is  divideo*  into  two  equal  parts,  the  individual  electricities  alone 
remain  in  each  part  respectively ;  but  in  the  case  of  magnetic 
induction,  although  no  appearance  of  polarity  be  exhibited  ex- 
cept at  the  two  ends,  yet  wherever  a  fracture  is  made,  the  two 
ends  separated  by  the  fracture  immediately  exhibit  opposite  po- 
larities, each  being  of  an  opposite  name  to  that  of  the  original 
pole  at  the  other  end  of  the  fragment.  If  each  of  the  two  frag- 
ments be  again  divided  into  any  number  of  parts,  each  of  these 
parts  is  a  magnet  perfect  in  itself,  having  two  opposite  poles. 

In  magnetism,  therefore,  there  is  never,  as  in  electricity,  any 
transfer  of  properties,  but  only  the  excitation  of  such  as  were 
already  inherent  in  the  body  acted  upon.  Magnetism  never 
passes  out  of  one  body  into  another ;  nor  can  we  ever  obtain  a 
piece  of  iron  or  steel,  that  contains  exclusively  either  northern  or 
southern  polarity. 

»  Christie,  Phil.  Trans.,  1825.  t  Barlow. 


MAGNETISM.  457 

678.  The  force  of  attraction,  or  of  repulsion,  exerted  upon  each 
other  by  the  poles  of  two  magnets,  placed  at  different  distances,  va- 
ries inversely  as  the  square  of  the  distance. 

This  law  was  ascertained  by  Coulomb,  by  means  of  the  tor- 
sion balance,  in  a  manner  similar  to  that  adopted  in  investiga- 
ting the  law  of  electrical  attraction.  (See  Art.  583.)  The  same 
law  therefore  which  governs  the  attraction  of  gravitation,  like- 
wise controls  electrical  and  magnetic  attractions.  It  is  the  most 
extensive  law  of  the  physical  world.  Nor  is  this  action  at  a 
distance  prevented,  or  even  impaired,  by  the  interposition  of  oth- 
er bodies  not  themselves  magnetic. 

679.  The  magnetic  power  of  iron  resides  wholly  on  its  SURFACE, 
and  is  independent  of  the  mass. 

Thus  a  hollow  globe  of  iron  of  a  given  surface  will  have  the 
same  effect  on  the  needle  as  though  it  were  solid  throughout.* 
In  this  fact  we  again  meet  with  a  striking  analogy  between  mag- 
netism and  electricity,  a  similar  property  having  before  been 
shown  to  belong  to  the  electric  fluid.  This  is  one  of  the  most 
recent  discoveries  in  magnetism,  and  was  made  by  Professor 
Barlow  of  the  Military  Academy  at  Woolwich,  (Eng.)  to  whose 
ingenious  and  assiduous  labors  are  due  many  of  the  latest  and 
most  important  investigations  in  this  science. 


CHAPTER  II. 

OF  THE  DIRECTIVE  PROPERTIES  OF  THE  MAGNET. 

680.  If  a  small  needle  be  placed  near  one  of  the  poles  of  a  mag- 
net, with  its  center  in  the  axis  of  the  magnet,  it  will  take  a  direction 
in  a  line  with  that  axis. 

Thus,  let  S  N  be  a  large  magnetic  bar,  and  s  n  a  small  needle 
placed  near  the  north  pole  of  the  magnet  with  its  center  in  the 
axis  ;  it  will  be  seen  that  the  action  of  the  pole  of  the  magnet 
is  such  as  to  bring  the  needle  into  a  line  with  the  magnet.  The 
action  of  the  bar  upon  the  needle  Fig.  229. 

tending  to  give  it  this  direction,  is, 
since  it  repels  n  and  attracts  s,     S          M          N 
equal  to  the   sum  of  its  actions 
upon  both  poles. 

681.  If  the  needle  be  placed  at  right  angles  to  the  bar,  with  one 
of  its  poles  directed  toward  the  center  of  the  bar,  it  will  take  a  di- 
rection parallel  to  the  bar. 

*  It  appears,  however,  that  a  certain  thickness  is  necessary  to  the  maximum  ef- 
feet,  although  that  thickness  is  very  small.     (See  Phil.  Trans.,  1831,  p.  81.) 
58 


458  NATURAL   PHILOSOPHY. 

By  supposing  B  (Fig.  229,)  to  be  placed  as  indicated  in  the 
above  proposition,  it  will  be  seen,  that  the  actions  of  both  poles 
of  the  magnet  would  conspire  in  relation  to  each  pole  of  the 
needle,  and  that  these  forces  can  be  in  equilibrium  only  when 
the  needle  is  parallel  with  the  bar.  The  needle  in  this  situation 
has  a  tendency  to  move  toward  the  magnet,  because  the  at- 
traction being  exerted  on  the  nearer,  and  the  repulsion  on  the 
remoter  pole,  the  sum  of  the  attractions  exceeds  that  of  the  re- 
pulsions. 

682.  Iron  filings  or  other  ferruginous  bodies,  which  are  free  to 
obey  the  action  of  a  magnetic  bar,  naturally  arrange  themselves  in 
curve  lines  from  one  pole  of  the  magnet  to  the  other. 

Thus,  if  we  place  a  sheet  of  white  paper  on  a  magnetic  bar 
laid  on  the  table,  and  sprinkle  iron  filings  on  the  paper,  the  fil- 
ings will  arrange  themselves  Fig<  230. 

in  curves  around  the  poles  of 
the  magnet.  A  small  sewing 
needle  suspended  horizontally 
by  a  slender  string,  on  being 
brought  near  to  different  parts 
of  the  magnet,  will  take  direc- 
tions corresponding  to  the  part  of  the  curve  in  which  it  happens 
to  be  placed.  At  the  poles  it  will  be  in  a  line  with  the  axis  of 
the  magnet  ;  opposite  the  center  of  the  bar  it  will  be  parallel  to 
it  ;  and  between  these  two  points  it  will  take  intermediate  di- 
rections, as  is  represented  in  Fig.  231. 

Fig.  231. 


These  curves  have  given  rise  to  the  most  fanciful  theories  of 
magnetism,  having  been  assumed  as  the  traces  of  an  invisible 
fluid  perpetually  circulating  between  the  poles  of  the  magnet  ; 
and  this  circulation  has  been  afterwards  employed  for  illustra- 
ting every  variety  of  magnetic  phenomena,  but  in  such  a  way 
as  to  leave  the  subject  involved  in  greater  mystery  than  at  first. 
These  causes  are  nothing  more  than  the  necessary  results  of 
forces  like  those  described  in  the  foregoing  propositions.* 

The  curves  which  iron  filings  describe  when  thus  arranged, 

*    Barlow. 


MAGNETISM.  459 

are  called  magnetic  curves.  They  present  several  curious  pro- 
perties, which  have  been  investigated  by  mathematicians ;  but 
we  must  refer  the  student  to  more  extensive  treatises  than  the 
present  for  a  full  development  of  this  subject.* 

683.  By  different  methods,  nearly  all  bodies  may  be  made  to  affect 
the  magnetic  needle. 

Coulomb,  by  the  assistance  of  the  extremely  delicate  appara- 
tus he  employed,  detected,  as  he  thought,  some  slight  portion  at 
least  of  magnetism  in  various  metals  besides  those  to  which  it 
had  been  exclusively  ascribed,  and  hence  announced  that  all  bodies 
whatsoever  are  subject  to  the  magnetic  influence.  The  possibility, 
however,  that  the  different  bodies  tried  by  Coulomb  might  con- 
tain small  portions  of  iron,  threw  some  doubt  on  his  conclusions  ; 
but  within  a  few  years  the  doctrine  has  been  confirmed,  that 
magnetism,  in  some  way  or  other,  pervades  all  bodies. 

M.  Arago,  of  France,  found  that  the  oscillations  which  a  nee- 
dle would  make  when  set  in  motion,  were  affected  by  its  being 
placed  above  or  below  plates  of  different  substances,  as  metal, 
ice,  or  water.  The  number  of  oscillations  in  a  given  time,  re- 
mained the  same  whatever  substance  was  employed  for  the 
plate  ;  but  the  horizontal  range  or  amplitude  of  the  needle  was 
much  greater  in  some  cases  than  in  others.  Copper,  for  example, 
would  diminish  the  range  four  times  as  much  as  lead,  and  a  hun- 
dred times  as  much  as  antimony.  This  fact  led  to  the  discovery, 
that  if  plates  of  copper  and  various  other  substances  are  put  in- 
to rapid  rotation  below  a  horizontal  needle  freely  suspended,  the 
needle  will  be  made  to  revolve  with  great  velocity.  The  exper- 
iment has  been  varied  in  many  different  ways.  Thus,  a  pow- 
erful horseshoe  magnet  itself,  placed  vertically,  has  been  made 
to  revolve,  having  a  plate  of  metal  or  other  substance  suspended 
immediately  over  it.  When  the  motion  of  the  magnet  acquires 
a  certain  velocity,  the  suspended  plate  will  likewise  revolve  in 
the  same  direction,  and  sometimes  with  so  great  a  speed  that  the 
eye  cannot  distinguish  it. 

Mr.  Faraday,  of  England,  has  recently  asserted  the  doctrine, 
supported  by  numerous  experiments,  that  all  the  metals  are  mag- 
netic at  a  certain  temperature,  and  that  at  a  certain  other  tem- 
perature, they  all  lose  this  power.  Even  iron  loses  all  magnetic 
properties  at  an  orange  heat,  and  nickel  at  a  heat  still  lower. 

684.  The  magnetic  needle,  when  freely  suspended,  seldom  points 
directly  to  the  pole  of  the  earth,  but  its  deviation  from  that  pole  is 
called  the  DECLINATION  or  the  VARIATION  of  the  needle. 

A  vertical  circle  drawn  through  the'  line  in  which  the  needle 
naturally  places  itself,  is  called  the  magnetic  meridian.  A  plane, 

*  Journal  of  the  Royal  Institution,  Feb.  1831— Leslie's  Geometrical  Analysis. 


460  NATURAL   PHILOSOPHY. 

passing  at  right  angles  to  the  magnetic  meridian,  through  the 
center  of  the  needle,  is  called  its  magnetic  equator.  A  line  drawn 
on  the  surface  of  the  earth,  passing  through  the  places  where 
the  needle  points  directly  to  the  north  pole,  and  where  of  course 
the  geographical  and  magnetic  meridians  coincide,  is  called  the 
line  of  no  variation. 

The  discovery  of  the  variation  of  the  needle  has  been  com- 
monly ascribed  to  Columbus.  His  son  Ferdinand  states,  that  on 
the  14th  of  September,  1492,  his  father  first  discovered  the  va- 
riation, and  that  in  consequence,  his  crew  mutinied,  supposing 
that  the  needle  had  lost  its  polarity,  and  that  they  would  not  be 
able  to  find  their  way  back  to  Europe.  It  appears,  however 
that  the  same  phenomenon  had  been  discovered  about  two  hun- 
dred years  before  that  period,  though  it  had  not  become  general- 
ly known  to  navigators.* 

685.  The  declination  of  the  needle  is  not  constant,  but  is  subject 
to  a  small  annual  change,  which  carries  it  to  a  certain  limit  on  one 
side  of  the  pole  of  the  earth,  when  it  becomes  stationary  for  a  time, 
and  then  returns  to  the  pole,  and  proceeds  to  a  certain  limit  on  the 
other  side  of  it,  occupying  a  period  of  many  years,  during  each  vi- 
bration. 

At  London,  in  the  year  1580,  the  needle  pointed  11£  degrees 
to  the  ea&t  of  north  ;  in  1657,  it  pointed  directly  to  the  pole  ;  af- 
ter which  period,  it  continued  to  move  west  for  one  hundred  and 
fifty-seven  years,  until  the  year  1814,  when  its  western  declina- 
tion was  nearly  24£  degrees ;  since  1814,  it  has  been  moving 
slowly  eastward.  If  it  takes  as  many  years  to  return  as  it  did 
to  move  from  the  pole  to  its  western  limit,  it  will  reach  the  pole 
again  in  the  year  1971  ;  and  should  it  proceed  as  far  eastward 
as  it  did  westward,  and  occupy  as  long  a  time,  it  will  reach  its 
eastern  limit  in  2128.  The  total  arc  of  declination  will  be  48° 
35'  48",  and  the  period  occupied  in  passing  over  it,  three  hun- 
dred and  fourteen  years.  This  would  be  an  average  of  9'  17" 
annually.  But  the  annual  variation  is  much  smaller  than  this 
toward  its  eastern  and  western  limits,  but  much  greater  when 
the  needle  is  in  the  vicinity  of  the  line  of  no  variation.  Thus, 
during  the  nine  years  that  elapsed  between  1814  and  1823,  the 
progress  eastward  was  only  1 1'  22",  or  only  1'  1.6"  annually,  while 
from  1657  to  1672,  a  period  of  fifteen  years,  the  declination  west 
amounted  to  2°  30',  or  10'  annually ;  and  between  1692  and  1722, 
the  annual  increase  of  declination  was  16' 40".  It  performed 
half  the  amount  of  its  western  declination  in  fifty-seven  years, 
while  to  complete  the  other  half,  occupied  one  hundred  years,  f 

The  variation  of  the  needle,  however,  is  not  the  same  at  the 


«  Cavallo,  Treatise  on  Magnetism,  Supplement— Barlow,  Phil.  Trans.  1833. 
t  Thomson,  Outlines  of  Heat,  Elec.,  and  Mag.,  p.  545. 


MAGNETISM.  461 

same  time  in  all  parts  of  the  earth,  but  every  part  has  its  par- 
ticular decimation.  For  instance,  if  we  sail  from  the  Straits  of 
Gibraltar  to  the  West  Indies,  in  proportion  as  we  recede  from 
Europe  and  approach  America,  the  compass  will  point  nearer 
and  nearer  due  north ;  and  when  we  draw  near  the  American 
coast,  it  will  point  exactly  north.  But  if  we  sail  from  Great 
Britain  to  the  southern  coast  of  Greenland,  we  shall  find  the  nee- 
dle deviate  further  and  further  from  the  north,  as  we  approach 
Greenland,  where  the  deviation  will  not  be  less  than  50g.*  In 
some  parts  of  Baffin's  Bay  the  needle  points  due  west. 

686.  The  line  of  no  variation  encompasses  the  globe,  but  its 
course  is  subject  to  numerous  irregularities.     Commencing  at 
the  magnetic  pole  in  Lat.  70°,  Lon.  90°  30',  it  runs  a  few  degrees 
east  of  south,  through  Hudson's  Bay,  New  South  Wales,  Lake 
Erie,  the  western  part  of  the  state  of  Pennsylvania,  passfcig  a 
little  westward  of  Washington  City,  and  enters  the  Atlantic 
Ocean  near  Newbern  in  North  Carolina.  After  leaving  the  United 
States,  it  veers  a  little  more  to  the  east,  running  a  few  degrees 
eastward  of  the  West  India  Islands,  and  meeting  the  eastern 
coast  of  South  America,  near  Maranham,  on  the  northeast  coast 
of  Brazil.      Crossing  the*  great   eastern  promontory  of  South 
America,  it  pursues  a  regular  course  in  a  southeasterly  direction 
toward  the  south  polar  regions.     In  the  eastern  hemisphere,  the 
line  of  no  variation  presents  greater  anomalies.     Proceeding 
from  south  to  north,  it  passes  in  a  northerly  direction  through 
New  Holland,  somewhat  westward  of  the  center.     Thence  its 
course  changes  to  the  west,  and  it  sweeps  in  a  great  curve 
through  the  Indian  Ocean,  around  Hindostan,  returning  through 
China  to  nearly  the  same  longitude  as  it  had  in  New  Holland, 
where  it  turns  to  the  north  again.     Places  lying  westward  of 
either  of  these  lines,  have,  within  certain  limits,  easterly  varia- 
tions, and  those  lying  eastward  of  those  lines,  have  westerly  va- 
riations.!    In  Europe,  Africa,  and  the  western  parts  of  Asia, 
together  with  the  greater  part  of  the  Atlantic  Ocean,  the  varia- 
tion is  to  the  west.     Throughout  the  greater  part  of  the  western 
hemisphere,  the  variation  is  to  the  east. 

687.  At  New  Haven,  the  variation  of  the  needle  is  at  present 
about  6  degrees,  and  is  on  the  increase.     In  1820,  it  was  4°  35' 
10"  ;$  and  in  1835,  it  was  5°  40'  34".  §     On  this  subject,  Profes- 
sor Loomis  remarks  :  From  the  time  of  the  earliest  observations, 
(1673,)  down  to  about  the  commencement  of  the  present  cen- 
tury, the  westerly  variation  was  decreasing,  and  the  easterly  in- 

*  Thomson,  Outlines  of  Heat,  Elec.,  and  Mag.,  p.  543. 

t  Barlow,  Phil.  Trans.,  1833. 

t  According  to  observations  made  by  Professor  Fisher.    See  Amer.  Jour.,  XVI.  60. 

§  Professor  Loomis,  Ib.  XXX,  224. 


462  NATURAL   PHILOSOPHY. 

creasing,  in  every  part  of  the  United  States.  Since  that  period, 
the  movement  of  the  needle  has  been  in  the  opposite  direction. 
At  present,  therefore,  the  westerly  variation  is  increasing,  and  the 
easterly  diminishing.  This  change  commenced  between  the 
years  1793  and  1819,  though  probably  not  everywhere  simulta- 
neously. The  present  annual  change  of  variation,  is  about  2 
minutes  in  the  southern  and  western  states,  from  3  to  4  minutes 
in  the  middle  states,  and  from  5  to  7  minutes  in  New  England. 
The  variation  for  several  prominent  points  is  as  follows : 


Cambridge,  Mass.        I*?? 
New  Haven,  Conn. 
New  York  City, 

-  in  1835, 
1836, 
1837, 
1836, 

8°  51'  \\ 
5°  55'  " 

5°  40'  " 
6°  47' 

Philadelphia, 
Washington  City, 
Dharlottesville,  Va. 
Newbern,  N.  C. 
Athens,  Geo. 
St.  Louis,  Mo. 

-       1837, 
-       1809, 
-       1835, 
-       1806, 
1837, 
1819, 

3°  52' 
0°  52' 
0°    0' 

2°    0'  E 
4°  31' 
10°  47' 

688.  Besides  the  annual  variation,  the  magnetic  needle  is  subject 
to  daily  changes  called  the  DIURNAL  VARIATION. 

According  to  the  observations  of  Professor  Loom  is,  made  at 
Yale  College  in  1835  and  1836,  the  north  end  of  the  needle  has 
in  the  morning  a  motion  eastward,  amounting  to  from  one  to 
three  minutes,  when  the  declination  is  usually  less  than  at  any 
other  hour  of  the  day,  and  may  therefore  be  called  the  minimum. 
During  winter,  this  minimum  is  attained  about  8  o'clock,  but  as 
early  as  7  o'clock  during  summer.  After  reaching  its  minimum 
position,  it  gradually  moves  to  the  west,  and  attains  its  maximum 
declination  about  3  o'clock  in  winter,  and  1  o'clock  in  summer. 
From  this  time  the  needle  again  returns  eastward.  The  whole 
amount  of  the  diurnal  variation  rarely  exceeds  12  minutes,  and 
is  commonly  much  less  than  that.  These  changes  of  declination 
during  the  day  are  connected  with  changes  of  temperature,  being 
from  May  to  October,  inclusive,  11'  56",  and  from  November  to 
April,  5'  11". 

689.  A  needle  first  balanced  horizontally  on  its  center  of  gravity 
and  then  magnetized,  no  longer  retains  its  level,  but  its  north  pole 
spontaneously  takes  a  direction  to  a  point  below  the  horizon,  catted 

the  DIP  OF  THE  NEEDLE. 

The  Dipping  Needle  is  represented  in  figure  232.  When  used, 
it  is  to  be  placed  in  the  magnetic  meridian,  and  the  stand  which 
supports  it  rendered  perfectly  level,  by  means  of  the  adjusting 
screws  attached. 


MAGNETISM.  463 

The  dip  of  the  needle  is  very  differ-  Fig.  232. 

ent  in  different  parts  of  the  globe  be- 
ing in  general  least  in  the  equatorial 
and  greatest  in  the  polar  regions.  At 
certain  places  on  the  globe  the  needle 
has  no  dip,  that  is,  it  becomes  perfect- 
ly horizontal,  and  a  line  uniting  all 
such  places  is  called  the  magnetic 
equator  of  the  earth.  Again  in  the 
polar  regions,  the  dipping  needle  some- 
times  becomes  nearly  perpendicular 
to  the  horizon.  In  the  middle  latitudes,  the  dip  is  greater  or 
less,  but  does  not  correspond  exactly  to  the  latitude. 

If  the  magnetic  meridian  coincided  with  the  geographical,  the 
magnetic  equator  would  coincide  with  the  earth's  equator ;  but 
such  is  not  the  fact.  We  may  consider  the  magnetic  equator,  in 
general,  as  a  great  circle  encompassing  the  earth,  and  inclined 
to  its  equator  at  an  angle  of  about  12  degrees.  It  not  only  cross- 
es the  equator  at  two  points  diametrically  opposite  to  each  other, 
as  a  regular  great  circle  would  do,  but  crosses  it  also  in  one  or 
perhaps  two  intermediate  points. 

The  dip  of  the  needle,  like  the  declination,  is  not  constant  at 
the  same  place,  but  undergoes  a  slight  variation  from  year  to 
year.  In  the  course  of  two  hundred  and  forty-five  years,  it  has 
varied  at  London  more  than  5°.  Its  present  amount  is  about 
70°,  and  the  variation  is  from  two  to  three  minutes  annually. 
The  following  table  exhibits  the  dip  of  the  magnetic  needle  for 
the  year  1839,  at  the  places  annexed.* 

Hudson,  Ohio, Lat.  41°  15'  N 72°  48'.4 

Buffalo, 42°  53'   74°  40'.8 

Schenectady, 42°  48'   74°  36M 

Albany, 42°  39'   74°  51'.3 

New  York  City, 40°  43'   72°  52;.2 

New  Haven,  Conn., 41°  18'   73°  26'.7 

Cambridge,  Mass., 42°  22'   74°  20'.  1 

Princeton,  N.  J., 40°  22'   72°  47'.1 

Washington  City 38°  53'  71°  21'.4 

690.  The  force  exerted  by  the  magnetism  of  the  earth  varies  in 
different  places :  its  comparative  estimate  for  any  given  place,  is 
called  the  MAGNETIC  INTENSITY  for  that  place. 

As  in  the  case  of  the  pendulum  in  its  relation  to  the  force  of 
gravity,  the  magnetic  intensity  may  be  measured  by  the  number 
of  oscillations  (Art.  183)  which  a  needle  drawn  a  given  number 
of  degrees  from  its  point  of  rest,  performs  in  a  certain  time,  as  a 
minute  for  example,  the  force  being  as  the  square  of  the  number 

*  Professor  Loomis,  Amer.  Phil.  Trs.,  1839. 


464  NATURAL   PHILOSOPHY. 

of  oscillations.  In  general  it  is  well  ascertained  that  the  mag- 
netic intensity  is  least  in  the  equatorial  regions,  and  increases  as 
we  advance  toward  the  poles.  It  is  probably  at  its  maximum 
at  the  magnetic  poles.  By  ascertaining,  from  actual  observa- 
tion, a  number  of  different  places  on  the  surface  of  the  earth, 
where  th  -agnetic  intensities  are  equal,  and  connecting  them 
by  a  line,  it  Appears  that  they  arrange  themselves  in  a  curve 
around  the  magnetic  pole.  These  lines  are  called  isodynamic 
curves.  Extensive  journeys  have  been  undertaken  by  Humboldt, 
Sabine,  Hansteen,  and  others,  to  ascertain  the  point  on  the  sur- 
face of  the  earth  where  the  magnetic  intensities  are  equal,  for 
the  purpose  of  describing  these  curves.  The  earlier  results  indi- 
cated the  position  of  the  magnetic  pole  to  be  in  the  northeastern 
part  of  Hudson's  Bay,  lat.  60°  N.,  Ion.  80°  W.  ;*  but  the  direc- 
tions of  these  curves  presented  such  anomalies  as  to  suggest  the 
idea  of  a  second  magnetic  pole  in  the  opposite  hemisphere  :  with 
the  view  of  ascertaining  this  point,  Professor  Hansteen,  of  Chris- 
tiana, several  years  since,  undertook  a  journey  into  Siberia,  at 
the  expense  of  the  King  of  Sweden,  and  has  fully  confirmed  the 
fact,  that  there  exists  a  second  magnetic  pole  to  the  north  of  Si- 
beria, around  which  the  isodynamic  curves  arrange  themselves 
in  regular  order,  f  From  experiments  made  in  deep  mines,  and 
in  the  upper  -regions  of  the  atmosphere  by  aeronauts,  it  appears, 
that  in  both  these  situations,  the  magnetic  intensity  is  the  same 
as  at  the  corresponding  places  on  the  surface  of  the  earth. 

691.  The  effects  produced  by  the  earth  on  a  magnetic  needle,  cor- 
respond to  those  produced  on  it  by  a  powerful  magnet,  and  hence 
the  earth  itself  may  be  considered  as  such  a  magnet. 

The  magnetism  of  the  earth  has  been  supposed  by  some  to 
result  from  a  great  magnet  lying  in  the  central  parts  of  the 
earth  ;J  by  others,§  to  be  nothing  more  than  the  resultant  of  all 
the  smaller  magnetic  forces  scattered  through  various  parts  of 
the  terrestrial  sphere  ;  and  by  others,  to  be  excited  on  the  sur- 
face of  the  earth  by  the  action  of  the  solar  rays. 

The  supposition  of  a  great  magnet  in  the  interior  of  the  earth, 
to  which  all  the  phenomena  of  terrestrial  magnetism  are  to  be 
ascribed,  is  the  earliest  hypothesis,  and  is  adequate  to  explain 
most  of  the  facts  of  the  science.  But  such  a  supposition  is  in- 
consistent with  the  recent  discovery  of  two  north  poles,  (Art. 
690,)  implying  the  existence  of  four  magnetic  poles  of  the  earth. 

*  In  the  year  1832,  Commander  James  Ross,  of  the  British  navy,  supposed  that  he 
had  reached  the  true  magnetic  pole  in  N.  lat.  70°,  W.  Ion.  96°  30',  in  a  region  lying 
northward  of  Hudson's  Bay,  and  westward  of  Baffin's  Bay.  The  hest  chart  hith- 
erto published  of  the  magnetic  lines  of  equal  variation  on  the  earth's  surface,  has 
recently  been  constructed  by  Professor  Barlow,  from  an  immense  number  of  observa- 
tions, and  published  in  the  Philosophical  Transactions  for  1833. 

t  Sabine,  Amer.  Jour.  XVII,  145.  $  Gilbert. 

§  Humboldt  and  Biot 


MAGNETISM.  465 

The  opinion  of  Biot,  that  terrestrial  magnetism  is  only  the  ag- 
gregate, or  resultant,  of  all  the  individual  magnetic  forces  resid- 
ing in  different  parts  of  the  earth,  appears  to  be  no  improbable 
supposition,  and  accords  well  with  the  general  doctrine  of  the 
composition  of  forces. 

602.  In  the  year  1813,  Dr.  Morichini,  of  Rome,  announced  that 
the  violet  rays  of  the  solar  spectrum  have  the  property  of  ren* 
dering  iron  magnetic.  In  1825,  these  experiments  were  repeated 
and  extended  by  Mrs.  Somerville,*  and  resulted  in  proving,  that 
the  magnetizing  power  is  not  confined  to  the  violet  rays,  but  ex- 
tends to  the  indigo,  blue,  and  green  rays.  The  probable  conclu- 
sion is,  that  a  class  of  rays  emanate  from  the  sun,  which  have  the 
property  of  producing  magnetism,  and  are  distinct  from  those 
which  afford  light  and  heat,  and  produce  chemical  changes. 
Hence,  in  the  solar  beam  there  are  at  least  four  distinct  kinds  of 
rays,  denominated,  respectively,  colorific,  calorific,  chemical  and 
magnetizing  rays.f 

693.  Electricity  and  magnetism  are,  in  some  of  their  properties, 
remarkably  alike,  but  in  others  strikingly  dissimilar. 

Several  of  these  analogies  have  been  already  incidentally  men- 
tioned ;  but  it  will  be  useful  to  the  student  to  consider  them  in 
connection.  Electricity  and  magnetism  agree  in  the  following 
particulars.  (1.)  Each  consists  of  two  species,  the  vitreous  and 
resinous  electricities,  and  the  austral  and  boreal  magnetisms. 
(2.)  In  both  cases,  those  of  the  same  name  repel,  and  those  of 
opposite  names  attract  each  other.  (3.)  The  laws  of  induction 
in  both  are  very  analogous.  (4.)  The  force,  in  each,  varies  in- 
versely as  the  square  of  the  distance.  (5.)  The  power,  in  both 
cases,  resides  at  the  surface  of  bodies,  and  is  independent  of 
their  mass. 

But  electricity  and  magnetism  are  as  remarkably  unlike  in  the 
following  particulars.  (!'.)  Electricity  is  capable  of  being  excited 
in  all  bodies  and  of  being  imparted  to  all :  magnetism  resides  al- 
most exclusively  in  iron  in  its  different  forms,  and,  with  a  few  ex- 
ceptions, cannot  be  excited  in  any  other  than  ferruginous  bodies. 
(2'.)  Electricity  may  be  transferred  from  one  body  to  another-: 
magnetism  is  incapable  of  such  transference  ;  magnets  commu- 
nicate their  properties  merely  by  induction,  a  process  in  which 
no  portion  of  the  fluid  is  withdrawn  from  the  magnetizing  body. 
(3'.)  When  a  body  of  an  elongated  figure  is  electrified  by  induc- 
tion, on  being  divided  near  the  middle,  the  two  parts  possess  re- 
spectively the  kind  of  electricity  only  which  each  had  before  the 
separation  ;  but  when  a  bar  of  steel  or  a  needle  magnetized  by 
induction,  is  broken  into  any  number  of  parts,  each  part  has  both 
polarities  and  becomes  a  perfect  magnet.  (4'.)  The  directive 

*  Phil.  Trans.  1826.  t  See  Brewster's  Optics,  p.  92. 

59 


466  NATURAL   PHILOSOPHY. 

properties  and  the  various  consequences  that  result  from  it,  the 
decimation,  annual  and  diurnal  variations,  the  dip,  and  the  differ- 
ent intensities  in  different  parts  of  the  earth,  are  all  peculiar  to 
the  magnet,  and  do  not  appertain  to  electrified  frodies. 

694.  The  phenomena  of  magnetism  are  explained  on  the  hypo- 
thesis of  two  fluids,  residing  naturally  in  iron  and  all  ferruginous 
bodies,  which,  when  united,  neutralize  each  other's  effects,  but  which, 
when  separated,  exhibit  the  respective  properties  of  boreal  and  aus- 
tral magnetism. 

Nearly  all  the  arguments  alleged  in  favor  of  the  hypothesis  of 
two  fluids  in  electricity,  apply  equally  well  to  magnetism.  It  is 
necessary  however  to  assume,  that  the  two  magnetic  fluids  are 
separated  from  each  other  only  at  distances  extremely  small,  for 
otherwise  it  is  impossible  to  account  for  the  fact,  that  when  a  mag- 
net is  divided  into  minute  fragments,  each  piece  contains  both 
fluids,  being  a  perfect  magnet  with  two  opposite  poles.  This  hy- 
pothesis, like  the  corresponding  one  in  electricity,  has  been  sub- 
mitted by  Poisson  to  the  most  rigorous  mathematical  analysis, 
and  all  the  deductions  made  from  it  are  found  to  accord  exactly 
with  the  facts  ascertained  by  experiment.  Hence  this  doctrine  is 
generally  received,  and  has  nearly  superseded  the  explanation 
formerly  given  by  ^Epinus,  who  accounted  for  magnetic  phenom- 
ena on  the  supposition  of  a  single  fluid,  similar  to  Franklin's  hy- 
pothesis of  electricity. 

According  to  the  foregoing  hypothesis,  iron  differs  from  nearly 
every  other  natural  substance,  in  containing  a  certain  portion  of 
the  compound  magnetic  fluid.  This  usually  maintains  its  equi- 
librium, and  therefore  is  latent  or  insensible  ;  but  various  causes 
disturb  this  state  of  equilibrium,  and  then  the  separate  fluids  ex- 
hibit their  peculiar  properties.  When  once  separated,  they  have 
the  power  of  producing  on  the  magnetic  fluid  of  other  masses  of 
iron  a  similar  separation,  each  repelling  the  similar,  and  attracting 
the  dissimilar  species.  Hence  one  magnet  affords  the  means  of 
making  another  ;  and  the  process  of  magnetizing  consists  not  in 
imparting  any  thing  from  the  magnetizing  body,  but  merely  in 
decomposing  the  fluid  before  residing  in  the  body  magnetized,  that 
is,  separating  it  into  its  constituent  fluids.  Indeed,  so  far  from 
losing  by  the  process  of  magnetizing,  the  original  magnet  itself 
gains  by  the  reaction  of  the  new  magnet  it  has  formed,  which 
tends  still  more  fully  to  develope  or  separate  its  own  constituent 
fluids.  By  this  means,  what  was  originally  a  very  weak,  may 
become  a  strong  and  powerful  magnet,  without  any  other  aid, 
than  what  contributes  to  separate  more  fully  the  two  fluids  natu- 
rally inherent  in  it.* 


*  Pouillet,  Etemens  de  Physique,  I,  325 


MAGNETISM.  467 

695.  The  facility  with  which  soft  iron  acquires  and  loses  mag- 
netism, (Art.  675,)  is  conceived  to  depend  on  the  ease  with  which 
the  magnetic  fluids  pervade  a  mass  of  loose  texture,  in  which  the 
particles  have  comparative  freedom  of  motion  ;  while  the  greater 
fixedness  of  the  particles  of  hard  steel,  creates  an  obstruction  to 
the  motions  of  the  same  fluids.     Thus  a  magnet  loses  its  powers 
by  exposure  to  a  white  heat,  (Art.  676,)  because  the  separate  flu- 
ids, having  freedom  of  motion,  combine  and  neutralize  each  other ; 
and  the  method  of  imparting  magnetism  to  iron  by  magnetizing 
it  while  softened  by  heat  and  suddenly  cooling  it,  is  so  effectual, 
because  in  this  way  the  two  fluids  are  first  easily  separated  by  in- 
duction, and  afterward  are  prevented  from  combining  by  the  in- 
creased obstruction  created  by  hardening  the  metal.     The  devel- 
opment of  magnetism  in  an  iron  bar  by  percussion,  (Art.  676,) 
is  supposed  to  be  owing  likewise  to  the  greater  freedom  of  motion 
secured  to  the  magnetic  fluids  by  the  vibration  of  the  particles  of 
iron,  thus  enabling  these  fluids  to  separate  from  one  another,  while 
as  soon  as  the  vibration  ceases,  that  freedom  of  motion  is  lost, 
and  the  fluids  are  prevented  from  reuniting.     That  such  a  vi- 
bration is  favorable  to  the  effect  produced,  is  inferred  from  the 
fact  that  blows  which  produce  a  ringing  sound  are  peculiarly 
efficacious  in  developing  magnetism.     The  same  explanation  is 
applied  to  the  case  where  magnetism  is  lost  by  percussion  ;  since 
here,  the  vibrations  would  enable  the  separate  fluids  to  combine. 

The  periodical  changes  in  the  situation  of  the  magnetic  poles 
of  the  earth,  upon  which  the  direction  of  the  needle  depends,  in- 
cluding the  annual  and  diurnal  variations,  the  dip,  and  the  in- 
tensity of  the  force,  result  from  causes  which  have  hitherto 
eluded  discovery. 

METHODS    OF   MAKING   ARTIFICIAL   MAGNETS. 

696.  If  the  learner  has  made  himself  acquainted  with  the 
principles  expounded  in  Jhe  preceding  propositions,  he  will  be 
qualified  to  proceed,  with  interest  and  intelligence,  to  an  expla- 
nation of  the  leading  methods  practised  in  the  manufacture  of 
artificial  magnets.     These  methods  also,  by  involving  a  practi- 
cal application  of  those  principles,  will  serve  to  impress  them  on 
the  memory  and  to  render  the  knowledge  of  them  familiar. 

It  will  be  recollected  that  magnets  are  made  from  other  mag- 
nets ;  that  this  is  done,  not  by  any  transference  of  a  portion  of 
the  power  of  the  magnetizing  body,  but  by  the  development  of 
the  powers  naturally  residing  in  the  body  to  be  magnetized  ;  that 
this  development  is  effected  wholly  on  the  principle  of  induction  ; 
that  the  original  magnet  gains  instead  of  losing  by  its  action  on 
other  bodies ;  that  this  power  may  be  induced  on  iron  by  the 
agency  of  an  artificial  magnet,  or  of  the  loadstone,  or  of  the 
earth,  which  is  itself  a  weak  magnet,  and  acts  upon  the  same 


468  NATURAL    PHILOSOPHY. 

principles  as  any  other  magnet.  It  must  also  be  kept  clearly  in 
mind,  that  soft  iron  or  steel  readily  acquires  and  as  readily  loses 
the  magnetism  induced  upon  it,  and  that  hardened  iron  or  steel 
.receives  it  slowly  and  with  much  difficulty,  but  retains  it  per- 
manently. As  the  earth  itself  may  be  supposed  to  have  been 
the  original  source  of  magnetism  in  all  other  bodies  in  which  it 
is  found,  we  shall  begin  by  describing  the  methods  of  magnetiz- 
ing from  the  earth,  without  the  aid  of  either  a  loadstone  or  an 
artificial  magnet. 

697.  A  certain  degree  of  magnetism  may  be  given  to  steel  bars 
by  hammering  them  while  in  a  vertical  position. 

Bars  of  steel  prepared  for  this  purpose  are  of  a  prismatic  form 
with  rectangular  sides,  the  length  being  ten  times  the  breadth, 
and  twenty  times  the  thickness.  Six  or  eight  bars  of  equal  size 
are  to  be  provided,  and  being  held  in  a  vertical  position  they  are 
to  be  struck  with  a  few  blows  of  the  hammer,  when  they  will 
be  found  to  have  acquired  a  feeble  degree  of  magnetism,  which 
is  indicated  by  their  exhibiting  polarity  and  having  the  power  of 
attracting  iron  filings.  This  effect  will  be  much  greater  if  the 
bars,  while  receiving  the  blows,  be  placed  near  to  a  mass  of  iron, 
so  as  to  experience  its  inductive  influence.  A  pair  of  tongs  may 
be  used  for  this  purpose  ;  during  the  process,  the  tongs  them- 
selves become  magnetic,  and  by  their  reaction  greatly  increase 
the  magnetism  of  the  bars. 

698.  A  needle  may  be  magnetized  by  simply  suffering  it  to  re- 
main in  contact  with  the  pole  of  a  strong  magnet ;  or  better,  be- 
tween the  opposite  poles  of  two  magnets. 

The  effect  produced  by  two  magnets  is  much  more  than  double 
that  of  one  magnet,  as  may  be  inferred  from  article  673.  But  if 
the  needle  be  of  considerable  length,  several  intermediate  sets  of 
poles  are  sometimes  developed,  as  will  be  seen  by  applying  iron 
filings.  It  adds  much  to  the  power  of  the  two  magnetic  bars 
between  which  the  needle  is  placed,  if  to  the  extremity  of  the 
bar  most  remote  from  the  needle,  a  mass  of  soft  iron  is  placed. 
(See  Art.  673,)  The  iron,  in  this  case,  acts  and  reacts  by  induc- 
tion :  and  hence  whenever  magnets  are  not  in  use,  they  require 
to  be  connected  with  iron  to  prevent  the  loss  of  their  powers. 
Pieces  of  soft  iron  thus  connected  with  magnets  for  the  purpose 
of  augmenting  their  powers  by  induction,  are  called  armatures. 
Thus  A  is  the  armature  of  the  horse-shoe  magnet  represented 
in  Fig.  234. 

699.  But  it  must  be  recollected,  (Art.  693,)  that  the  two  spe- 
cies of  magnetism  are  not,  like  those  of  electricity,  separated  to 
a  distance  from  each  other,  so  that  one  kind  may  be  wholly  col- 
lected at  one  end  of  the  bar  and  the  other  kind  at  the  other  end, 


MAGNETISM.  469 


but  that  the  two  are  separated  only  at  a  minute  distance,  re- 
maining in  the  immediate  vicinity  of  each  other  throughout  the 
whole  length  of  the  bar.  Hence,  in  order  to  give  the  magnetiz- 
ing pole  its  full  effect,  it  becomes  necessary  to  apply  it  succes- 
sively to  every  part  of  the  bar  from  one  end  to  the  other. 

A  more  effectual  method  of  magnetizing  a  needle  is  the  fol- 
lowing : — Place  two  magnetizing  bars,  A,  B,  parallel  to  each 
other,  with  their  dissimilar  poles  adjacent ;  unite  the  poles  at  one 
end  by  a  piece  of  soft  iron,  R,  and  apply  the  poles  at  the  other 
end  to  the  needle,  as  is  represented  in  figure  233.  Upon  this 
principle,  that  is,  the  increased  energy  with  which  the  two  poles 

Fig.  234. 


act  together,  is  formed  what  is  called  the  horse-shoe  magnet, 
which  derives  its  name  from  its  peculiar  figure,  (Fig.  234.)  Bars 
of  this  form  are  converted  into  magnets  upon  the  same  prinfi- 
ples  as  straight  bars,  the  magnetizing  bar  being  made  to  follow 
the  curvature  always  in  the  same  direction.  A  very  efficacious 
mode  of  making  horse-shoe  magnets  is  thus  described  by  Profes- 
sor Barlow.  Two  horse-shoe  bars  may  be  united  at  their  ends 
in  such  a  manner,  that  the  poles  which  are  to  be  of  opposite  names 
shall  be  in  contact.  They  are  then  to  be  rubbed  with  another 
strong  horse-shoe  magnet,  placing  the  latter  so  that  its  north 
pole  is  next  to  the  south  pole  of  one  of  the  new  magnets,  and 
consequently  its  south  pole  next  to  the  north  pole  of  the  same  ; 
carrying  the  movable  magnet  round  and  round  always  in  the 
same  direction.  This  is  esteemed  one  of  the  most  eligible  modes 
of  making  powerful  magnets. 

The  horse-shoe  magnet  is  itself  very  convenient  for  imparting 
magnetism  to  other  bodies.  Place  the  poles  near  the  center  of 
the  needle  ;  move  them  along  its  surface  backward  and  forward, 
taking  care  to  pass  over  each  half  of  it  an  equal  number  of 
times ;  repeat  the  same  operation  on  the  other  side ;  and  the 
needle  will  become  speedily  and  effectually  magnetized. 

700.  The  best  mode  of  making  magnetic  needles  in  general, 
is  expressed  in  the  following  rule,  given,  as  the  result  of  very  ex- 
tensive and  accurate  experiments,  by  Capt.  Kater. 

Place  the  needle  in  the  magnetic  meridian ;  join  the  opposite 
poles  of  a  pair  of  bar  magnets,  (the  magnets  being  in  the  same  line,) 
and  lay  the  magnets  so  joined  jlat  upon  the  needle,  with  their  poles 
upon  its  center ;  then  having  elevated  the  distant  extremities  of  the 


470  NATURAL   PHILOBOPHV. 

magnets,  so  that  they  may  form  an  angle  of  about  two  or  three  de- 
grees with  the  needle,  draw  them  from  the  center  of  the  needle  to  the 
extremities,  carefully  preserving  the  same  inclination  ;  and  having 
joined  the  poles  of  the  magnets  at  a  distance  from  the  needle,  repeat 
the  operation  ten  or  twelve  times  on  each  surface.* 

In  connection  with  the  foregoing  rule,  Capt.  Kater  gives  the 
following  summary  of  principles,  established  with  respect  to  the 
compass  needle.  1.  That  the  best  material  for  compass  needles 
is  a  clock  spring ;  but  care  must  be  taken,  in  forming  the  needle, 
to  expose  it  as  seldom  as  possible  to  heat,  otherwise  its  capability 
of  receiving  magnetism  will  be  much  diminished.  2.  That  the 
best  form  of  a  compass  needle  is  a  pierced  rhombus,  (Fig.  236,) 
in  the  proportion  of  about  five  inches  in  length  to  two  in  width, 
this  form  being  found  susceptible  of  the  greatest  directive  force. 
3.  That  the  best  method  for  tempering,  is  first  to  harden  the  nee- 
dle at  a  red  heat,  and  then  to  soften  it  from  the  middle  to  about 
an  inch  from  each  extremity,  by  exposing  it  to  heat  sufficient  to 
cause  the  blue  color  which  arises,  again  to  disappear.  4.  That 
in  the  same  plate  of  steel,  of  the  size  of  a  few  square  inches  only, 
portions  are  found  varying  considerably  in  their  capability  of  re- 
ceiving magnetism,  though  not  apparently  differing  in  any  other 
r^pect.  5.  That  polishing  the  needle  has  no  apparent  effect 
on,  its  magnetism.  6.  That  in  needles  from  five  to  eight  inches 
in  length,  their  weights  being  equal,  the  directive  forces  are  nearly 
as  the  'lengths.  7.  That  the  directive  force  does  not  depend  upon 
extent  of  surface,  but,  in  needles  of  the  same  length  and  form, 
it  is  as  the  mass.  8.  The  deviation  of  a  compass  needle,  occa- 
sioned by  the  attraction  of  soft  iron,  depends  on  extent  of  surface 
and  is  wholly  independent  of  the  mass,  except  a  certain  thickness 
of  iron,  amounting  to  about  two  tenths  of  an  inch,  which  is  re- 
quisite for  the  complete  development  of  its  attractive  energy. 

701.  The  reasons  on  which  the  preceding  rule  and  the  annexed 
principles  are  founded,  will  for  the  most  part  be  understood  from 
what  has  gone  before.  A  needle  to  be  magnetized  is  placed  in 
the  magnetic  meridian,  because  (the  earth  being  considered  as  a 
magnet)  the  needle  has  its  axis  then  parallel  to  that  of  the  mag- 
net, a  position  in  which,  Art.  671,  it  receives  the  greatest  effect 
from  induction.  The  opposite  poles  are  joined,  because  acting 
reciprocally  upon  each  other  by  induction,  they  augment  each 
other's  powers.  The  bars  thus  joined  are  placed  on  the  center  of 
the  needle  and  drawn  in  opposite  directions,  for,  by  this  means, 
upon  that  part  of  the  needle  which  lies  between  them,  the  forces 
of  the  two  poles  conspire.  Upon  the  part  which  lies  between 
each  bar  and  the  adjacent  extremity  of  the  needle,  these  two 
poles  are  opposed  to  each  other ;  but  as  the  poles  are  more  remote 

*  Phil.  Trans. 


MAGNETISM.  471 

from  the  parts  where  their  actions  are  opposed  to  each  other,  than 
from  those  whose  actions  conspire,  they  on  the  whole  tend  to 
augment  each  other's  effects.  The  bars  are  first  laid  flatwise,  and 
afterward  elevated  by  as  small  an  angle  as  will  serve  the  pur- 
pose of  drawing  them  asunder,  with  their  poles  only  in  contact 
with  the  needle,  because  the  effect  of  induction  is  strongest 
when  the  magnetizing  bars  are  nearest  to  a  parallelism  with  the 
body  to  be  magnetized ;  and  the  same  angle  of  inclination  is 
carefully  preserved/ for  it  is  only  in  this  way  that  both  sides  of 
the  needle  will  have  precisely  the  same  strength,  a  condition  es- 
sential to  its  perfection.  In  renewing  the  application  of  the 
bars,  they  are  removed  to  a  distance  before  their  poles  are  joined 
again,  because  it  is  important  to  secure  the  magnetism  which 
the  needle  has  already  acquired,  against  those  partial  disturb- 
ances which  might  arise  from  the  irregular  action  of  the  mag- 
netic bar. 

702.  Magnets  are  liable  to  lose  their  power,  to  prevent,  which, 
certain  precautions  are  necessary. 

If  a  single  magnet  is  kept  out  of  its  natural  direction,  it  grows 
gradually  weaker,  and  this  loss  of  power  is  most  rapid  when  its 
position  is  the  reverse  of  the  natural  one,  that  is,  when  its  north 
pole  is  turned  toward  the  south.  Under  these  circumstances, 
indeed,  unless  the  magnet  is  made  of  the  hardest  steel,  it  will  in 
no  long  time  lose  the  whole  of  its  magnetic  power.  Two  mag- 
nets may  also  very  much  weaken  each  other  if  they  are  kept, 
even  for  a  short  time,  with  their  similar  poles  fronting  each  other. 
The  polarity  of  the  weaker  magnet,  especially,  is  rapidly  im- 
paired, and  sometimes  found  to  be  actually  reversed.  More  fre- 
quently, however,  there  arises  from  this  opposition  of  powers, 
considerable  irregularity  and  confusion  in  the  poles  of  both  mag- 
nets. 

Since  heat  also  impairs  the  powers  of  the  magnet,  (Art.  675,) 
this  instrument  should  never  be  exposed  to  a  high  temperature. 
We  should  likewise  be  very  cautious  to  avoid  all  rough  and  vio- 
.  lent  treatment ;  for  its  virtues  are  speedily  impaired  by  concussion, 
or  whatever  occasions  a  vibration  among  its  particles.  It  must 
not,  therefore,  be  suffered  to  fall  on  the  floor,  or  be  rubbed  with 
coarse  powders,  or  be  ground  with  the  view  of  altering  its  form. 
The  loadstone  has  its  powers  impaired  by  similar  means  ;  hence, 
we  should  attempt  to  alter  its  natural  form  as  little  as  possible ; 
and  when  it  is  necessary  to  do  so,  it  should  be  effected  very  rap- 
idly by  cutting  it  on  a  lapidary's  wheel. 

Although  the  loadstone  retains  its  magnetic  virtue  more  tena- 
ciously than  any  artificial  magnet  that  can  be  constructed,  yet 
even  this  body  requires  a  certain  management  for  the  permanent 
preservation  of  its  powers.  For  this  purpose  it  should  be  armed, 
as  it  is  called  ;  that  is,  a  piece  of  soft  iron  should  be  constantly 


472 


NATURAL   PHILOSOPHY. 


kept  in  contact  with  the  two  poles.     In  order  to  do  this  most  ef- 

fectually, we  must  first  ascertain  the  situation  of  the  poles  of  the 

loadstone,  which  we  may  do  by  rolling  it  in  iron  filings  ;  and  then 

cutting  off  all  the  superfluous  parts,  we 

may  give  it  the  shape  of  a  parallele- 

piped, having  the  poles  in  the  middle  of 

two  opposite  surfaces,  and  at  the  same 

time  taking  care  to  preserve  the  axis, 

which  passes  through  the  poles,  of  as 

great  a  length  as  can  be  obtained  ;  for 

it  has  been  observed  that  any  curtail- 

ment of  the  magnet  in  the  direction  of 

this  line,  deprives  it  of  force  in  a  greater 

degree  than  when  shortened  in  any  oth- 

er direction.     Plates  of  soft  iron   are 

next  attached  to  the  two  sides  contain- 

ing the  poles,  which  are  made  to  pro- 

ject a  little  way  below  the  bottom  of  the  loadstone,  terminating 

in  two  bars,  like  the  poles  of  a  horse-shoe  magnet,  to  which  bars 

a  short  bar  of  soft  iron  is  attached,  upon  which  the  whole  force 

of  both  poles  acts  simultaneously.     This  action  exerted  upon  the 

iron  bar,  is  sufficient  to  preserve  the  powers  of  the  loadstone 

from  decay.     (See  Fig.  235.)     A  similar  piece  of  iron  is  applied 

by  way  of  armature,  to  the  two  poles  of  a  horse-shoe  magnet. 

Bar  magnets  also,  when  laid  aside,  should  be  placed  with  the 

north  pole  of  one  in  contact  with  the  south  pole  of  another,  or 

what  is  better*  two  bars  may  be  placed  parallel,  at  a  little  dis- 

tance from  each  other,  with  their  like  poles  in  opposite  directions, 

and  having  their  dissimilar  poles  united  by  short  pieces  of  iron, 

so  as  to  form,  with  the  bars,  a  parallelogram.     Magnets  should 

be  polished,  because  they  are  then  less  liable  to  contract  rust.* 

703.  The  compass  (the  importance  of  which  to  mankind,  has 
attached  to  the  subject  of  magnetism  its  principal  value)  is  of 
many  different  forms,  but  the  chief  varieties  are  the  land  com- 
pass, the  Mariner's  compass,  the  Azimuth  compass,  and  the  Va- 
riation compass.  The  needle,  in  all  these  varieties,  is  usually  a 
thin  flat  plate  of  steel,  tapering  at  the  extremities  ;  but,  as  we 
have  already  mentioned,  (Art.  700,)  a  more  eligible  form  has 
been  proposed  by  Capt.  Kater,  consisting  of  four  narrow  strips 
of  steel,  united  in  the  form  of  a  hollow  rhombus,  (Fig.  236.)  It  is 
found  advantageous  to  concentrate  F'ls-  236- 

the  powers  of  the  needle  as  much 
as  possible  in  the  two  extremities, 
and  to  avoid  all  inequalities,  aris- 
ing from  intermediate  poles,  or 


Lib  Useful  Knowledge,  Magnetism,  p.  54. 


MAGNETISM.  473 

from  a  difference  of  strength  in  different  parts.  The  needle  is 
secured  at  the  point  of  suspension,  and  furnished  with  a  conical 
cap  of  brass  which  rests  on  a  perpendicular  pin  ;  and  still  fur- 
ther to  diminish  friction,  the  point  which  rests  on  the  extremity 
of  the  pin  is  made  of  agate,  one  of  the  hardest  mineral  sub- 
stances. Since,  if  the  needle  is  magnetized  after  having  been 
balanced  on  its  center  of  gravity,  it  would  no  longer  remain 
horizontal,  the  equipoise  is  restored  by  attaching  a  small  weight 
to  the  elevated  side. 

704.  The  compass,  in  its  simplest  form,  consists  of  a  needle  like 
the  foregoing,  enclosed  in  a  suitable  box  covered  with  glass. 
This  is -all  that  is  essential  when  it  is  required  merely  to  know 
the  direction  of  the  meridian,  or  the  north  and  south  points. 
But,  for  most  purposes,  the  compass  is  furnished  with  a  graduat- 
ed circular  card,  divided  into  degrees  and  minutes  ;  and  in  the 
mariner's  compass,  the  card  is  also  divided  into  thirty-two  equal 
parts,  called  rhumbs.     The  card  thus  divided  is  fastened  to  the 
needle  itself,  and  turns  with  it. 

Thin,  slender  needles,  have  the  greatest  directive  powers,  and 
are  most  sensible,  since  they  undergo  less  friction  than  those 
which  are  heavier,  but  due  regard  to  strength  requires  them  to 
be  made  of  a  certain  degree  of  thickness  ;  an  increase  of  length 
is  attended  with  an  increase  of  directive  power ;  but  when  the 
thickness  remains  the  same,  the  weight,  and  consequently  the 
friction,  increases  in  the  very  same  ratio  ;  no  advantage,  there- 
fore, as  to  directive  power,  can  be  obtained  by  any  increase  of 
length.  Moreover,  needles  which  exceed  a  very  moderate 
length,  are  liable  to  have  several  sets  of  poles,  a  circumstance 
which  is  attended  with  a  great  diminution  of  directive  force. 
On  this  account,  short  needles,  made  exceedingly  hard,  are  gene- 
rally preferable. 

/ 

705.  The  great  importance  of  the  Mariner's   compass,  has 
made  its  construction  an  object  of  much  attention,  and  the  best 
artists  have  tried  their  skill  upon  it.     The  compass  is  suspended 
in  its  box  in  such  a  manner  as  to  remain  in  a  horizontal  position, 
notwithstanding  all  the  motions  of  the  ship.     This  is  effected  by 
means  of  gimbals.     This  contrivance  consists  of  a  hoop,  CD, 
usually  of  brass,  (Fig.  237,)  fastened  horizontally  to  the  box  by 
two  pivots  placed  opposite  to  each  other,  and  constituting  the 
axis  on  which  the  hoop  turns  up  and  down.     At  an  equal  dis- 
tance from  the  pivots  on  each  side,  that  is,  at  the  distance  of  90° 
from  each  pivot,  two  other  pivots  are  attached  to  the  Ting  at 
right  angles  to  the  former,  on  which  the  inner  box  that  contains 
a  card  is  huug.     Of  course  when  it  turns  on  these  pivots,  its  mo- 
tion is  at  right  angles  with  that  of  the  hoop.     Therefore  all  the 
motions  of  which  the  compass  box  is  capable,  are  performed 

60 


474 


around  two  axes  which  intersect  each  other  at  right  angles  ;  con- 
sequently, the  point  of  intersection,  being  in  both  axes,  will  not 
move  at  all.  But  the  needle  and  the  attached  card  rest  upon 
this  point,  and  are  connected  with  the  compass  box  in  no  other 
point.  Hence  they  remain  constantly  horizontal  in  every  posi- 
tion of  the  box. 

The  Azimuth  compass*  differs  from  the  common  mariner's 
compass  only  in  having  sights  attached,  by  which  the  bearing  of 
any  object  with  the  meridian  may  be  ascertained.  The  Survey- 
or's compass  is  a  variety  of  the  azimuth  compass. 

LOCAL   ATTRACTION    OF   VESSELS. 

706.  A  few  years  since  it  was  observed,  for  the  first  time,  that 
the  needle  of  the  mariner's  compass,  on  board  of  a  ship,  does  not 
continue  to  point  constantly  in  the  same  direction,  but  alters  its 
direction  as  the  ship  heads  toward  different  parts.  Changing  the 
position  of  the  ship  from  north  or  south  to  east  or  west,  some- 
times changes  the  direction  of  the  needle  20°  or  30°.  Indeed,  in 
one  instance  mentioned  by  Capt.  Sabine,  shifting  the  ship's  head 
from  east  to  west,  produced  a  change  in  the  direction  of  the  nee- 
dle amounting  to  50°.  Such  irregularities  are  found  to  be  great- 
est in  the  polar  seas.  This  effect  is  caused  by  the  attraction 
which  the  large  quantity  of  iron  on  board  a  ship  exerts  upon  the 

*  Azimuth,  as  applied  to  a  star  or  any  celestial  object,  is  an  arc  of  the  horizon  in- 
tercepted between  the  meridian  and  a  vertical  circle  passing  through  the  object. 


MAGNETISM.  475 

needle,  consisting  of  the  guns  on  board  of  a  man  of  war,  of  the 
masses  of  iron  sometimes  employed  as  ballast,  of  the  iron  tanks 
recently  substituted  for  water  casks,  and  of  the  various  bolts, 
bars,  nails,  &c.  which  enter  more  or  less  into  the  construction  of 
every  sort  of  vessel.  Indeed,  on  account  of  the  greater  quantity 
of  iron  now  employed,  the  mariner's  compass,  at  present,  is  more 
subject  to  irregularities  from  this  cause  than  it  was  formerly.* 

In  order  to  investigate  the  laws  by  which  these  effects  were 
controlled,  and  to  devise  a  remedy  for  them,  Professor  Barlow,  of 
the  Military  Academy  of  Woolwich,  instituted  a  great  number  of 
experiments,  which  resulted  in  the  discovery. of  a  method  of  ob- 
viating completely  every  difficulty,  by  neutralizing  the  effect  of 
the  iron  of  the  ship,  and  leaving  the  needle  free  to  obey  the  im- 
pulse of  terrestrial  magnetism  alone.  It  is  easy  to  understand 
that  all  the  various  forces  exerted  by  the  iron  in  different  parts 
of  the  vessel,  will  have  a  single  resultant,  equivalent  to  the  whole  ; 
and  that,  if  we  can  discover  the  amount  of  this  resultant,  it  will 
be  only  necessary  to  make  a  suitable  correction,  to  be  either 
added  or  subtracted,  according  as  the  indications  of  the  needle 
are  too  small  or  too  great. 

707.  Mr.  Barlow  procured  a  solid  ball  of  iron,  thirteen  inches 
in  diameter,  and  two  hundred  and  eighteen  pounds  in  weight. 
When  the  compass  was  above  the  ball,  he  found  that  the  north 
end  of  the  needle  was  attracted  toward  it ;  and  that  when  it 
was  below  the  ball,  the  south  end  was  attracted  toward  it :  and 
that,  in  traversing  the  interval  between  these  two  positions,  it  al- 
ways passed  through  a  point  in  which  the  ball  had  no  effect  on 
the  needle.  Instead,  however,  of  moving  the  compass  through 
these  different  positions,  the  compass  was  suffered  to  remain  sta- 
tionary, and  the  ball,  suspended  by  means  of  pulleys,  was  raised 
or  lowered  at  pleasure,  and  thus  easily  brought  into  any  required 
position  with  respect  to  the  compass.  The  experiment  showed, 
that  all  those  points  in  which  the  ball  exerts  no  influence  on  the 
needle,  are  in  the  same  plane,  and  that  this  plane  is  inclined  to 
the  horizon  toward  the  south,  making  an  angle  with  the  horizon 
equal  to  the  complement  of  the  dip  ;  and  of  course  the  direction 
of  this  plane  is  at  right  angles  to  the  direction  of  the  dip.  This 
plane,  therefore,  in  reference  to  the  iron  sphere,  constitutes  its 
magnetic  equator.  It  is  at  right  angles  to  the  magnetic  meridian, 
and  cuts  the  horizon  in  the  magnetic  east  and  west  points.  A 
compass  needle,  whose  center  is  anywhere  in  this  plane,  will 
not  have  its  action  disturbed  in  the  least  by  the  influence  of  the 
ball.  Hence,  this  plane  is  denominated  the  plane  of  no  attraction, 
or  the  plane  of  neutrality.  Nor  is  the  existence  of  such  a  plane 
confined  to  masses  of  iron  of  a  globular  shape ;  it  extends  equal- 

*  Barlow,  Phil.  Trans.  1833. 


476  NATURAL   PHILOSOPHY. 

ly  to  masses  of  the  most  irregular  form,  and  even  to  an  assem- 
blage of  detached  masses,  like  those  disposed  through  the  differ- 
ent parts  of  a  ship. 

708.  The  actual  amount  of  deviation  produced  in  the  ship's 
compass  by  its  local  attraction,  will,  of  course,  be  different  in  dif- 
ferent vessels.     With  an  easterly  or  westerly  course,  it  has  been 
observed  in  the  latitude  of  London  to  vary  from  five  to  twelve  or 
fourteen  degrees  :  it  is  of  greater  amount  as  the  ship  is  in  higher 
latitudes ;  and  diminishes,  without  vanishing,  at  the  equator ; 
and  again  increases  as  we  approach  the  south  pole.     Experi- 
ments were  made  on  eight  different  men  of  war  in  the  British 
harbors,  and  in  all  of  them  very  considerable  deviations  were 
detected  from  the  local  cause  under  consideration,  and  an  average 
deviation  in  the  whole  of  8°  44'.     The  Gloucester,  one  of  these 
ships,  was  invariably  drawn  to  the  southward  of  her  intended 
place,  notwithstanding  the  greatest  care  was  taken  in  steering 
her.     Had  it  not  been  ascertained,  by  taking  an  observation,  that 
this  error  was  altogether  the  effect  of  local  attraction,  it  would 
probably  have  been  ascribed  to  the  influence  of  an  unknown 
current.     The  real  deviation,  estimated  in  distance,  would  occa- 
sion the  vessel,  after  running  ten  miles,  to  be  more  than  a  mile 
and  a  half  to  the  southward  of  her  reckoning,  and  so  on  as  the 
distance  increased.     An  error  of  this  magnitude,  occurring  in  a 
narrow  channel,  and  in  a  dark  night,  were  it  unknown  or  disre- 
garded, might  lead  to  the  most  disastrous  consequences ;  and 
shipwrecks  have  been  traced,  with  much  probability,  to  this 
source  of  error  in  the  reckoning.     The  loss  of  the  Thames,  India- 
man,  a  few  years  since,  was  ascribed  to  this  cause.     This  vessel, 
besides  the  usual  supply  of  guns,  had  a  cargo  of  more  than  four 
hundred  tons  of  iron  and  steel.     The  influence  of  such  an  enor- 
mous magnetic  mass,  would  alone  be  quite  sufficient  to  explain 
the  otherwise  unaccountable  circumstances,  that  after  leaving 
Beachey  Head  in  sight  at  six  o'clock  in  the  evening,  the  ship 
was  wrecked  upon  the  same  spot,  between  one  and  two  o'clock 
in  the  morning,  without  the  least  apprehension  of  being  near  the 
shore.* 

709.  The  Correcting  Plate  of  Professor  Barlow,  affords  an  ef- 
fectual remedy  for  these  errors.     It  consists  of  a  double  plate, 
formed  of  two  thin  disks  of  iron,  screwed  together  in  such  a 
manner  as  to  combine  any  strong  irregular  power  of  one  plate, 
with  a  corresponding  weak  part  of  another ;  by  which  means  a 
more  uniform  action  is  obtained.     These  plates  are  of  a  circular 
form,  twelve  or  thirteen  inches  in  diameter.     Now,  it  being  as- 
certained from  actual  experiment,  (comparing  the  direction  of  the 

*  Barlow. 


MAGNETISM.  477 

compass  on  board  with  its  direction  on  the  shore,)  what  is  the 
amount  of  deviation  occasioned  by  the  iron  of  the  ship,  it  is  evi- 
dently possible,  by  bringing  a  small  quantity  of  iron  near  to  the 
needle,  to  produce  in  it  a  deviation  of  the  same  amount,  and  of 
course  to  double  the  error  in  question.  In  Fig.  238,  Fig.  238. 
A  represents  a  vertical  stand  or  log  of  wood,  turning 
horizontally  on  its  base,  to  the  top  of  which  the 
compass  B  is  firmly  attached.  The  correcting  plate 
C,  is  supported  by  a  pin  passing  through  its  center, 
and  entering  a  hole  made  in  the  side  of  the  stand. 
Of  these  holes  there  are  several,  so  as  to  admit  of 
shifting  the  positions  of  the  plate.  To  ascertain 
the  local  attraction  of  a  ship,  the  direction  of  the 
needle  is  first  observed  on  board,  free  from  the 
influence  of  the  plate.  The  apparatus  represent- 
ed in  Fig.  238,  is  then  removed  on  shore,  and 
the  bearing  of  the  compass  observed  before  apply- 
ing the  plate.  The  difference  in  direction  on  board 
and  on  shore,  shows  the  effect  of  the  ship's  iron. 
The  plate  is  now  applied,  and  the  log,  together, 
with  the  plate,  is  turned  round,  until  the  direction 
of  the  needle  is  the  same  as  on  board.  In  order  to  make  it  the 
same,  it  may  be  necessary  to  shift  the  position  of  the  plate,  by 
inserting  the  pin  in  a  different  place.  We  have  now  ascer- 
tained the  position  of  the  plate,  with  respect  to  the  needle,  re- 
quired in  order  to  indicate  the  local  attraction  of  the  ship.  If 
the  apparatus  is  placed  again  on  board,  (the  compass  and  the 
plate  retaining  the  same  relative  position,)  the  whole  deviation 
of  the  needle  from  its  true  place  will  be  doubled ;  that  occa- 
sioned by  the  iron  of  the  ship  being  equal  in  amount  to  that  made 
by  the  plate.  Hence,  in  any  given  case,  we  have  only  to  observe 
the  effect  of  the  plate  upon  the  needle,  in  order  to  learn  the 
amount  of  the  local  attraction. 

In  order  to  bring  the  efficacy  of  the  correcting  plate  to  the 
test  of  experience,  several  of  the  ships  of  the  royal  navy  of 
Great  Britain  were  furnished  with  it,  and  trials  were  instituted 
with  it  in  various  parts  of  the  world,  from  the  arctic  to  the  ant- 
arctic circle,  and  with  the  most  satisfactory  results.  This  expe- 
dient, therefore,  is  at  present  held  to  be  a  most  effectual  correc- 
tive of  the  errors  from  the  local  attraction  of  vessels. 

710.  Chronometers,  also,  which  are  carried  on  board  of  ships 
for  the  purpose  of  finding  the  longitude,  are  liable  to  have  their 
rate  of  going  affected  by  the  magnetic  action  of  the  iron  of  the 
ship.  Although  a  sudden  alteration  in  the  rates  of  chronometers 
at  sea  had  frequently  been  observed,  yet  the  cause  was  not  de- 
tected until  as  late  as  the  year  1818.  It  appeared  on  examina- 
tion, that  the  effect  was  produced  by  the  magnetic  action  of  the 


478  NATURAL   PHILOSOPHY. 

ship's  iron  upon  the  inner  rim  of  the  balance  of  the  chronometer, 
which  is  made  of  steel.  A  similar  influence  was  perceptible  on 
placing  magnets  in  the  neighborhood  of  the  chronometer.  Mr. 
Barlow  applied  himself  to  experiments  on  this  subject,  and  found 
that  masses  of  iron  wholly  destitute  of  permanent  magnetism, 
occasioned  an  alteration  in  the  rates  of  chronometers,  placed 
near  them  in  different  positions.  Sometimes  they  wrere  accelera- 
ted, and  sometimes  retarded.  Hence,  it  is  recommended  to  keep 
the  chronometer,  on  board  of  any  ship,  out  of  the  vicinity  of 
any  large  mass  or  surface  of  iron.  The  method  proposed  for 
rectifying  this  error  is  the  same  as  that  for  correcting  the  com- 
pass, viz.  by  first  ascertaining  what  the  effect  of  the  ship's  iron 
upon  the  chronometer  is,  and  then  applying  the  correcting  plate 
upon  the  same  principles  as  in  the  case  of  the  compass. 

The  late  voyages  to  the  northern  seas,  undertaken  by  the  Brit- 
ish government,  however  they  may  have  failed  of  gaining  their 
principal  object,  namely,  the  discovery  of  a  northwest  passage, 
still  achieved  many  valuable  results  in  experimental  science,  but 
in  none,  perhaps,  more  than  in  the  science  of  magnetism.  Among 
the  rest,  they  made  numerous  observations  on  the  local  attrac- 
tion of  vessels ;  on  the  magnetic  effect  of  the  ship's  iron  upon 
the  rate  of  chronometers ;  upon  the  position  of  the  magnetic 
poles ;  upon  the  phenomena  of  the  dipping  needle ;  and  upon 
the  magnetic  intensities  of  different  places  on  the  earth's  sur- 
face. 


MAGNETIC  CHARTS. 

711.  The  great  importance  of  the  mariner's  compass  to  the 
art  of  navigation,  has  induced  the  British  government,  at  differ- 
ent times,  to  send  abroad  men  of  science  to  make  observations 
on  magnetism  in  different  countries,  with  the  view  of  reducing 
the  principles  on  which  the  variation  of  the  compass  depends,»to 
settled  laws.  The  first  great  enterprise  of  this  kind  was  under- 
taken about  the  year  1680,  by  Dr.Halley,  one  of  the  most  distin- 
guished and  zealous  philosophers  of  that  age.  For  the  purpose 
of  ascertaining  the  law  of  the  variation  of  the  compass,  Dr.  Hal- 
ley  was  invested  with  the  command  of  a  national  ship,  in  which 
he  traversed  the  Atlantic  ocean  in  various  parts,  extending  his 
voyage  to  the  fiftieth  degree  of  south  latitude.  After  he  had 
collected  a  great  number  of  observations  made  by  others,  and 
compared  them  with  his  own,  he  published,  in  the  year  1700,  a 
synopsis  of  them  in  the  form  of  a  chart,  in  which  the  ocean  was 
represented  as  crossed  by  a  number  of  lines  passing  through 
those  places  where  the  compass  had  the  same  deviation.  Thus, 
in  every  point  of  one  line  there  was,  in  the  year  1701,  no  vari- 
ation ;  in  any  point  of  another  line,  the  compass  had  twenty  de« 


MAGNETISM.  479 

grees  of  east  variation ;  and  in  every  point  of  a  third  line,  it  had 
twenty  degrees  of  west  variation.* 

But  though  Halley's  chart  was  constructed  with  all  possible 
care,  and  presented  a  comprehensive  view  of  all  that  was  then 
known  of  the  subject,  yet  it  could  not  be  of  much  permanent 
utility,  since  the  lines  of  which  it  is  composed  are  themselves 
continually  changing  their  relation  to  one  another.  Among  the 
recent  magnetic  charts  which  have  been  published,  that  of  Pro- 
fessor Barlow  is  the  most  extensive  and  useful,  f  Professor 
Loomis  has  published  a  valuable  chart  of  the  United  States,  in 
the  39th  volume  of  the  American  Journal  of  Science. 

The  great  and  constant  irregularities  of  all  the  lines  described 
on  magnetic  charts,  whether  they  relate  to  the  variation  of  the 
compass,  or  to  the  magnetic  dip  and  intensity,  are  such  as  almost 
to  preclude  the  hope  of  reducing  the  phenomena  of  terrestrial 
magnetism  to  laws  so  definite,  as  to  afford  rules  of  calculating 
these  particulars  for  any  given  place,  independently  of  experi- 
ment. 

712.  The  presence  of  the  Aurora  Boreatts,  is  found  to  have  a 
remarkable  effect  on  the  magnetic  needle.  A  brilliant  Aurora 
will  sometimes  cause  an  enormous  deflexion  in  the  needle.  In- 
deed this  deflexion  has  been  found,  in  certain  cases,  to  remain  to 
a  greater  or  less  degree  permanent.  In  the  great  Aurora  of  No- 
vember 14th,  1837,  the  needle  often  moved  thirty  minutes  in 
three  seconds,  and  its  entire  range  of  motion  was  no  less  than 
six  degrees.J  The  luminous  columns,  also,  which  frequently 
appear  during  this  phenomenon  are  generally,  if  not  always,  par- 
allel to  the  magnetic  meridian  ;§  and  the  Corona,  or  luminous 
circle,  which,  in  some  great  exhibitions  of  the  Aurora,  is  formed 
southeast  of  the  zenith,  has  its  center  at  the  point  toward  which 
the  upper  end  of  the  dipping-needle  is  directed,  that  is,  toward 
the  pole  of  the  dipping-needle.  These  facts  evidently  show  that 
the  matter  that  forms  the  Aurora  has  magnetic  properties,  but 
they  give  no  information  respecting  its  origin.  This  is  believed 
to  be  independent  of  the  earth,  and  the  discussion  of  it  therefore 
belongs  to  astronomy. 

»  Robison's  Mech.  Phil.  IV.,  358.  t  Phil.  Transactions  for  1833. 

J  Mr.  E.  C.  Herrick.  §  See  Dalton's  Met.  Essays.— Am.  Jour,  xxx,  227. 


48(>  NATURAL   PHILOSOPHY. 


PART  VIII. OPTICS. 

% 

CHAPTER  I. 

PRELIMINARY  DEFINITIONS  AND  OBSERVATIONS. 

713.  OPTICS  is  that  branch  of  Natural  Philosophy  which  treats 
of  Light  and  Vision. 

More  particularly,  it  is  the  object  of  this  science  to  investigate 
the  nature  of  the  agent  on  which  the  phenomena  of  vision  de- 
pend ;  to  treat  of  the  motions  of  light  in  respect  to  its  direction, 
its  velocity,  and  its  reflexion  from  the  surface  of  bodies  ;  to  trace 
its  change  of  direction,  and  the  various  other  modifications  it 
undergoes  by  passing  through  different  transparent  media ;  to 
explain  the  phenomena  of  nature  which  depend  upon  the  proper- 
ties of  light,  embracing  the  doctrine  of  color ;  to  trace  the  rela- 
tion between  light  and  the  structure  of  the  eye,  comprehending 
the  subject  of  vision  ;  and  finally,  to  describe  the  various  instru- 
ments to  which  a  knowledge  of  the  principles  of  Optics  has 
given  birth,  disclosing  many  new  and  wonderful  properties  of 
light,  and  extending  the  range  of  human  vision,  on  the  one  hand, 
to  myriads  of  objects  too  minute,  and  on  the  other,  to  number- 
less worlds  too  remote,  to  be  seen  by  the  unassisted  eye. 

714.  Luminous  bodies  are  naturally  of  two  kinds,  such  as  shine 
by  their  own  light,  as  a  lamp  or  the  sun,  and  such  as  shine  by 
borrowed  light,  as  the  moon,  and  most  of  the  visible  objects  in 
nature. 

A  ray  is  a  line  of  light ;  or  it  is  the  line  which  may  be  con- 
ceived to  be  described  by  a  particle  of  light.  In  a  more  general 
sense,  the  term  is  applied  to  denote  the  smallest  portion  of  light 
which  can  be  separately  subjected  to  experiment.  A  beam  is  a 
collection  of  parallel  rays.  A  pencil  is  a  collection  of  converging 
or  diverging  rays.  A  medium  is  any  space  through  which  light 
passes.  When  a  space  is  a  perfect  void,  so  as  to  offer  no  obstruc- 
tion to  the  passage  of  light,  it  is  said  to  be  a,  free  medium ;  when 
the  space  intercepts  a  portion  only  of  the  light,  it  constitutes  a 
transparent  medium.  Transparency,  however,  may  exist  in  dif- 
ferent degrees.  When  the  medium  itself  is  invisible,  as  portions 
of  air,  it  is  said  to  be  perfectly  transparent ;  when  the  medium 
is  visible,  but  objects  are  seen  distinctly  through  it,  as  in  the 


OPTICS.  481 

clearest  specimens  of  glass  and  crystal,  it  is  said  to  be,  simply, 
transparent ;  when  objects  are  indistinctly  seen  through  it,  it  is 
semi-transparent ;  and  when  a  mere  glimmering  of  light  passes 
through,  without  representing  the  figure  of  objects,  it  is  trans- 
lucent. Bodies  that  transmit  no  light  are  said  to  be  opake. 

715.  Rays  of  light,  while  they  continue  in  the  same  uniform  me- 
dium, proceed  in  straight  lines. 

For  objects  cannot  be  seen  through  bent  tubes ;  the  shadows 
of  bodies  are  terminated  by  straight  lines ;  and  all  conclusions 
drawn  from  this  supposition,  are  found  by  experience  to  be  true. 
If  two  bodies  with  plane  surfaces,  as  two  disks  of  metal,  be  held 
between  the  eye  and  some  luminous  point,  as'a  star,  on  bringing 
the  two  planes  gradually  toward  each  other,  the  star  may  be 
seen  through  the  intervening  space  until  the  planes  come  com- 
pletely into  contact ;  but  if  one  of  the  surfaces  is  convex  and  the 
other  concave,  the  light  is  intercepted  before  the  surfaces  have 
met.*  In  consequence  of  the  rectilinear  motion  of  light,  it  forms 
angles,  triangles,  cylinders,  cones,  dec.,  and  thus  its  affections  fall 
within  the  province  of  geometry,  the  principles  of  which  are 
applied  with  great  effect  to  the  development  of  the  properties 
and  laws  of  light,  after  a  few  fundamental  properties  are  estab- 
lished by  experiment. 

From  every  point  in  a  luminous  object,  an  inconceivable 
number  of  rays  of  light  emanate  in  every  direction,  when  not 
prevented  by  obstacles  that  intercept  it.  Thus,  from  every  point 
in  the  flame  of  a  candle,  as  seen  by  night,  light  diffuses  itself, 
pervading  an  immense  sphere,  and  filling  every  part  of  the  space 
so  perfectly,  that  not  the  minutest  point  can  be  found  destitute  of 
some  portion  of  its  rays.  Any  luminous  body  of  this  kind  is 
called  a  radiant.  The  pencil  of  light  which  proceeds  from  a 
radiant  is  a  cone,  the  sections  of  which  made  by  any  plane  cor- 
respond to  the  figures  called  conic  sections.  If  any  portion  of 
the  pencil  be  intercepted  by  a  rectilateral  figure,  that  portion 
constitutes  a  pyramid  of  which  the  figure  is  the  base,  and  the 
luminous  point  itself  is  the  vertex. 

716.  Light  has  a  progressive  motion  of  about  one  hundred  and 
ninety-two  thousand  Jive  hundred  miles  per  second. 

The  estimation  of  the  velocity  of  light,  (which  may  be  classed 
among  the  greatest  achievements  of  the  human  mind,)  has  been 
effected  in  two  different  ways.  The  first  method  is  by  means 
of  the  eclipses  of  Jupiter's  satellites.  To  render  this  mode  in- 
telligible to  those  who  have  not  studied  astronomy,  it  may  be 
premised,  that  the  planet  Jupiter  is  attended  by  four  moons  which 
revolve  about  their  primary,  as  our  moon  revolves  about  the  earth. 

»  Biot. 
61 


482  NATURAL    PHILOSOPHY. 

These  small  bodies  are  observed,  by  the  telescope,  to  undergo 
frequent  eclipses  by  falling  into  the  shadow  which  the  planet 
casts  in  a  direction  opposite  to  the  sun.  The  exact  moment 
when  the  satellite  passes  into  the  shadow,  or  comes  out  of  it,  as 
would  be  seen  by  a  spectator  at  the  mean  distance  of  the  earth 
from  the  sun,  is  calculated  by  astronomers.  But  sometimes  the 
earth  and  Jupiter  are  on  the  same  side,  and  sometimes  on  oppo- 
site sides  of  the  sun ;  consequently,  the  earth  is,  in  the  former 
case,  the  whole  diameter  of  its  orbit,  or  about  one  hundred  and 
ninety  millions  of  miles  nearer  to  Jupiter  than  in  the  latter. 
Now  it  is  found  by  observation,  that  an  eclipse  of  one  of  the 
satellites  is  seen  about  sixteen  minutes  and  a  half  sooner  when 
the  earth  is  nearest  to  Jupiter,  than  when  it  is  most  remote  from 
it,  and  consequently,  the  light  must  occupy  this  time  in  passing 
through  the  diameter  of  the  earth's  orbit,  and  must  therefore 
travel  at  the  rate  of  about  one  hundred  and  ninety-two  thousand 
miles  per  second.*  Another  method  of  estimating  the  velocity 
of  light,  wholly  independent  of  the  preceding,  is  derived  from 
what  is  called  the  aberration  of  the  fixed  stars.  The  full  expla- 
nation of  this  method  must  be  referred  to  astronomy ;  but  it  may 
be  understood  in  general,  that  the  apparent  place  of  a  fixed  star 
is  altered  from  the  effect  of  the  motion  of  its  light,  combined 
with  the  motion  of  the  earth  in  its  orbit.  It  will  be  remarked 
that  the  place  of  a  luminous  object  is  determined  by  the  direc- 
tion in  which  its  light  meets  the  eye.  But  in  the  case  of  light 
coming  from  the  stars,  the  apparent  direction  is  altered  in  conse- 
quence of  the  motion  of  the  earth  in  its  orbit,  being  intermediate 
between  the  actual  direction  of  the  earth  and  of  the  light  of  the 
star ;  and  the  velocity  of  the  earth  in  its  orbit  being  known,  that 
of  light  may  be  computed  from  the  proportional  part  of  the  effect 
produced  by  it  in  causing  the  aberration.  The  velocity  of  light, 
as  deduced  from  this  method,  comes  out  very  nearly  the  same 
as  by  the  other,  f  Hence  it  is  inferred,  that  the  velocity  of  light 
is  uniform. 

717.  The  intensity  of  light,  at  different  distances  from  the  radiant, 
varies  inversely  as  the  square  of  the  distance. 

This  proposition  is  proved  in  the  same  manner  as  that  respect- 
ing gravity,  (Art.  7,  p.  21,)  the  reasoning  in  which,  applies  to  all 
emanations  from  a  center. 

Although  the  intensity  of  light  decreases  rapidly  as  we  recede 
from  the  radiant,  yet  the  brightness  of  the  object  suffers  little  di- 
minution by  increase  of  distance.  Thus,  a  candle  appears 
nearly  as  bright  at  the  distance  of  a  mile  as  when  close  to  the 

*  190000000 

-—^ISSWOO  nearly. 

t  See  Herschel  on  Light,  Encyc.  Metrop. 


OPTICS.  483 

eye.  If,  while  the  observer  remains  stationary,  the  light  which 
was  before  spread  over  a  given  area,  should  be  all  collected  into 
a  space  half  as  large,  the  brightness  would  obviously  be  twice 
as  great  as  before  ;  or,  in  general,  the  brightness,  the  quantity 
of  light  being  given,  is  inversely  as  the  area ;  that  is,  inversely 
as  the  square  of  the  diameter.  Now,  as  we  recede  from  an  ob- 
ject, its  area  is  apparently  diminished,  and  on  this  account,  its 
brightness  is  increased  in  the  same  ratio  as  it  is  diminished  by  the 
cause  operating  according  to  the  foregoing  proposition.  The 
brightness  therefore  rem a' /inconstant.* 

This  is  to  be  understood  nowever,  only  of  light  passing  through 
a.  free  medium ;  by  traversing  the  air,  the  brightness  is  diminish- 
ed according  to  the  following  law. 

718.  The  effect  of  a  transparent  medium  of  uniform  density,  is 
to  diminish  the  intensity  of  light  in  a  geometrical  ratio. 

-For,  imagine  that  the  medium,  a  piece  of  glass,  for  example, 
is  divided  into  equal  laminae,  of  such  thickness,  that  the  first  la- 
mina shall  stop  -th  part  of  the  rays  that  fall  upon  it.  Then 

there  will  issue  from  the  lamina  1 = rays.  The  second 

n       n 

lamina,  in  like  manner,  will  stop  -th  part  of  the  light  which  falls 

n 

upon  it ;  that  is,  -  of  '"—L  =  11^-.   There  will,  therefore,  issue 

n—1     n—\       (n— I)2        T        , 

from  the  second  lamina, —  ___  =  v — — t-.       In    the    same 

n          ri*  v? 

manner  it  may  be  shown,  that  there  will  issue  from  the  third 

lamina,  ^n~  '  .      Hence,  the  series  expressing  the  decreasing 
n 

.    n-1     (n-1)'  (n—1)3    .  ,  .  ,    . 

quantities  of  light,  is ,  - — ^4 ,  i — x-^-,  &c.  which  is  evi- 

n          ws  n3 

dently  a  series  in  geometrical  progression.! 

719.  The  shadow  of  a  globe  that  is  illuminated  by  an  equal 
globe,  is  cylindrical,  and  indefinitely  long.     The  shadow  of  a 
less  globe,  illuminated  by  a  greater,  (as  of  the  earth,  or  of  the 
moon,  illuminated  by  the  sun,)  is  a  cone  of  finite  length,  whose 
dimensions  may  be  easily  computed  when  the  diameters  and  dis- 
tances of  the  globes  are  known.     And,  lastly,  the  shadow  of  a 
globe,  illuminated  by  one  that  is  smaller,  extends  itself  indefi- 

*  Herschel  on  Light.  t  Barlow. 


484 


NATURAL   PHILOSOPHY. 


nitely  in  a  truncated  cone,  perpetually  enlarging.    These  seve- 
ral truths  will  be  readily  understood  by  referring  to  Fig.  239. 

Fig.  239. 


Light,  when  it  impinges  on  smooth  surfaces,  is  reflected  back 
into  the  same  medium,  and  when  it  passes  out  of  one  medium 
into  another,  it  is  bent  out  of  its  former  course,  or  refracted.  The 
laws  of  reflexion  and  refraction  constitute,  severally,  important 
departments  of  the  science  of  Optics,  and  to  these  our  attention 
will  now  be  directed. 


CHAPTER  H. 
OF  THE  REFLEXION  OF  LIGHT  * 

720.  LIGHT  is  said  to  be  reflected  when,  on  impinging  upon  any 
surface,  it  is  turned  back  into  the  same  medium. 

Instruments  employed  as  reflectors  are  divided  into  mirrors 
and  speculums.  The  name  mirror  is  applied  to  reflectors  made 
of  glass  and  coated  with  quicksilver,  as  common  looking-glasses: 
the  word  speculum,  is  applied  to  a  metallic  reflector,  such  as  those 
made  of  silver,  steel,  tin,  or  of  a  peculiar  alloy,  called  speculum 
metal.  As  the  light  which  falls  on  glass  mirrors,  is  intercepted 
by  the  glass  before  it  is  reflected  from  the  quicksilvered  surface, 
a  speculum,  or  a  reflector  of  polished  metal,  is  that  supposed  to 
be  employed  in  optical  experiments,  unless  the  contrary  is  spe- 
cified. Such  a  surface,  indeed,  is  to  be  understood  where  the 
word  mirror  is  used  without  distinction. 

The  surface  of  the  mirror  or  speculum,  may  be  either  plane, 
concave,  or  convex,  and  the  reflector  is  denominated  accordingly. 

*  That  part  of  Optics  which  treats  of  reflected  light,  is  sometimes  denominated 
Catoptrics,  (Kor«J,)  and  that  part  which  treats  of  refracted  light,  Dioptrics,  &i*. 


OPTICS.  485 

A  ray  of  light  before  reflexion,  is 
called  the  incident  ray.  The  angle 
made  by  an  incident  ray,  at  the  sur- 
face of  the  reflector,  with  a  perpen- 
dicular to  that  surface,  is  called  the 
angle  of  incidence;  the  angle  made 
by  the  reflected  ray,  with  the  same 
perpendicular,  is  called  the  angle  of 
reflexion.  Thus,  in  Fig.  240,  if  MN 
represents  the  reflecting  surface,  DC  a  perpendicular  to  it  at  the 
point  C,  AC  the  incident,  and  BC  the  reflected  ray ;  then  ACD 
will  be  the  angle  of  incidence,  and  BCD  the  angle  of  reflexion. 

721.  Experiments  on  light  are  usually  conducted  in  a  room 
which  can  be  made  dark  with  close  shutters,  one  of  which  is 
perforated  with  a  circular  hole,  an  inch  or  two  in  diameter,  for 
admitting  a  beam  of  light.     This  opening  is  rendered  smaller  to 
any  required  degree  by  covering  it  with  a  piece  of  board  or  me- 
tallic sheet,  having  a  smaller  aperture.     And  as  the  sun  may  not 
shine  directly  into  the  shutter  at  the  time  required,  a  mirror  is 
sometimes  attached  to  the  outside  of  the  shutter,  so  contrived 
that,  by  means  of  adjusting  screws,  it  may  be  made  to  turn  the 
rays  of  the  sun  into  the  opening,  and  to  give  them  a  horizontal 
or  any  other  required  direction.     The  course  of  the  rays  is  ren- 
dered palpable  to  the  eye,  by  the  illuminated  particles  of  dust 
that  are  floating  in  the  air. 

722.  The  angles  of  incidence  and  reflexion  are  in  the  same  plane, 
and  are  equal  to  each  other. 

Let  a  ray  of  light  AC,  (Fig.  240,)  admitted  into  a  dark  cham- 
ber as  above,  be  incident  upon  a  horizontal  speculum  MN  at  the 
point  C,  to  which  the  line  CD  is  perpendicular,  and  let  CB  be  the 
reflected  ray.  Then  if  the  plane  surface  of  a  board  or  a  metallic 
plate,  be  made  to  coincide  with  the  incident  ray  and  the  perpen- 
dicular, it  will  be  found  to  coincide  also  with  the  reflected  ray, 
showing  that  the  three  rays  are  in  the  same  plane.  Again,  if, 
from  the  point  C,  with  the  radius  CA,  a  circle  be  described,  on 
measuring  the  arcs  subtended  by  the  angles  of  incidence  and  re- 
flexion, they  will  be  found  to  be  exactly  equal  to  each  other.* 
The  following  corollaries  will  be  evident  on  consideration  :  That 
the  complements  of  the  angles  of  incidence  and  reflexion,  are 
also  equal;  that  the  reflected  ray  may  be  taken  for  the  incident 
ray,  and  vice  versa  ;  and  that,  if  the  incident  ray  be  perpendicu- 
lar to  the  reflecting  surface,  it  will  be  reflected  back  in  the  same 

*  An  ingenious  apparatus  is  described  by  Blot,  (Precis  Elem.  tome  II,  136,)  by 
which  this  experiment  may  be  performed  with  the  utmost  degree  of  precision ;  the 
results  are  as  enunciated  in  the  proposition. 


486 


NATURAL   PHILOSOPHY. 


line.  The  angles  of  incidence  and  reflexion  are  also  equal  when 
the  reflexion  takes  place  from  a  concave  or  convex  surface  ;  for 
the  reflexion  being  from  a  point,  the  curve  and  tangent  plane 
at  that  point  coincide,  and  have  both  the  same  perpendicular, 
viz.  the  radius  of  the  curve. 

REFLEXION    OF   LIGHT   FROM   PLANE   MIRRORS. 

723.  When  rays  of  light  are  reflected  from  a  plane  surface,  the 
reflected  rays  have  the  same  inclination  to  one  another,  as  their  cor- 
responding incident  rays. 

Case  1.  Parallel  rays.— Let  RS  (Fig.  241,)  be  the  reflecting 
surface  ;  AB,  CD,  the  incident,  BG,  DH,  the  reflected  rays.  Then 
the  angle  CDR=ABR :  but  CDR=HDS,  and  ABR=GBS.  There- 
fore, HDS=GBS,  and  hence  BG  and  DH  are  parallel. 


Fig.  241. 


Fig.  242. 
A       j?      E 


R  D      B  8 

In  the  foregoing  example,  the  angles  of  incidence  are  sup- 
posed to  be  in  the  same  plane  ;  but  where  these  angles  .are  in 
different  planes,  let  AB,  CD,  (Fig.  242,)  be  two  parallel  rays  in- 
cident upon  the  plane  mirror  MNOP,  having  their  angles  of  in- 
cidence in  different  planes ;  from  their  points  of  incidence,  B,  D, 
draw  the  perpendiculars  BE,  DF,  and  let  BG,  in  the  plane  that 
passes  through  AB  and  BE,  be  the  reflected  ray ;  join  BD,  and 
let  DH  be  the  intersection  of  the  two  planes,  CDH  and  GBDH. 
Since  BE,  DF,  are  both  drawn  perpendicular  to  the  same  plane, 
they  are  parallel  ;*  and  as  AB  and  CD  are  parallel  by  supposi- 
tion, the  angles  of  incidence,  ABE,  CDF,  are  equal,  f  Because 
EB,  FD,  and  AB,  CD,  are  parallel,  the  planes  ABG,  CDH  are 
also  parallel,^  and  they  are  intersected  by  the  plane  GBDH  ;  con- 
sequently DH  is  parallel  to  BG.§  and  EBG=FDH.  But  EBG= 
ABE=CDF  /.  FDH=CDF,  and  hence  DH  is  the  reflected  ray, 
and  it  was  before  proved  to  be  parallel  to  BG,  the  other  reflect- 
ed ray. 

Case  2.  Diverging  Rays. — Let  RAB  (Fig.  243,)  be  a  pencil 
of  diverging  rays,  incident  upon  the  plane  mirror  PB  ;  and  from 
R  draw  RF  perpendicular  to  PAB,  and  cutting  the  mirror  in  P. 
Let  AD  be  the  reflexion  of  an  incident  ray  RA,  and  produce 


«  Euc.  6,  2,  Sup.        t  9,  2,  Sup.        {  13, 2,  Sup.        $  14,  2,  Sup. 


487 


DA  backward  to  F.  Then 
PAR=BAD=PAF;  conse- 
quently, in  the  right-angled 
triangles  PAR,  PAF,  the 
angles  are  all  equal,  and  PA 
common  ;  hence  RP  =  PF, 
that  is,  the  reflected  ray  pro- 
ceeds as  if  it  came  from  a 
point  F,  on  the  other  side  of 
the  mirror,  and  from  the  same 
distance  from  it  as  R.  In  like 
manner  it  may  be  shown, 
that  all  the  other  rays  will  proceed  as  if  they  diverged  from  F, 
and  therefore  F  is  the  virtual  focus,  or  imaginary  radiant,  of  all 
the  reflected  rays.  Since  PRA=PFA,  it  may  be  shown  in  the 
same  way  that  PRB=PFB  ;  hence,  taking  equals  from  equals, 
the  remainder  AFB=ARB,  that  is,  the  rays  after  reflexion  have 
the  same  inclination  as  before. 

Case  3.  Converging  Rays.— If  DA,  CB,  (Fig.  243,)  constitute 
a  pencil  of  incident  rays  converging  to  the  point  F,  it  follows 
from  the  above  reasoning  that  they  will  converge  to  the  focus  R 
after  reflexion. 

724.  Parallel  rays,  incident  upon  a  concave  mirror,  and  near  its 
axis,  are  reflected  to  a  focus  equidistant  from  the  surface  and  the 
center  of  the  mirror. 

Let  RA,  RE,  (Fig.  244,)  FiS-  244- 

be  parallel  rays  incident 
upon  the  spherical  mir- 
ror  AEB,  whose  center  is 
C.  The  ray  RE,  passing 
through  the  center  C,  and 
therefore  falling  perpen- 
dicularly on  the  mirror  at 
E,  will  be  reflected  in  the  direction  EC.  Having  joined  CA 
and  made  the  angle  CAF=CAR,  the  ray  RA  will  be  reflected  in 
the  direction  AF.  At  the  point  of  incidence  A,  draw  the  tan- 
gent GH,  cutting  CE  produced  in  H.  Then  because  RA  and 
RE  are  parallel,  the  angle  RAC=ACE=CAF  ;  consequently 
CF=FA.  But  since  CAH  and  CAG  are  equal,  and  likewise 
CAF  and  CAR,  .-.  FAH=RAG=FHA  .-.  FA=FH.  If  we  now 
suppose  the  ray  RA  to  approach  the  axis  RE,  the  arc  AE  will 
diminish,  and  its  secant  CH  will  ultimately  become  equal  to  the 
radius  CE,  and  then  FH  will  be  equal  to  FE,  and  of  course  FA 
or  FC  will  equal  FE. 

The  foregoing  proposition  is  applicable  to  such  rays  only  as 
are  exceedingly  near  to  the  axis  of  the  mirror  CE.  As  the  par- 
allel rays  are  more  remote  from  the  axis,  the  focus  F  approaches 


488  NATURAL   PHILOSOPHY. 

nearer  and  nearer  to  the  point  E,  until,  when  the  arc  EA  be- 
comes equal  to  60°,  F  coincides  with  E  ;  for  then  the  angle 
CAF  and  ACF  being  each  equal  to  60°,  the  remaining  angle  of 
the  triangle  ACF  must  also  be  equal  to  60°  ;  consequently,  CF 
must  equal  CA,  and  of  course  the  point  F  will  coincide  with  E. 

If  several  beams  of  parallel  rays  be  incident  nearly  perpendic- 
ularly upon  a  spherical  mirror,  the  foci  of  the  reflected  rays  will 
be  in  a  spherical  surface  concentric  with  that  mirror.  For  since 
each  focus  (Fig.  244)  is,  by  the  proposition,  equidistant  from  the 
center  and  from  the  surface  of  the  mirror,  the  distances  of  all 
the  foci  from  the  mirror  must  be  exactly  the  same  ;  that  is,  they 
must  be  in  a  surface  concentric  with  that  of  the  mirror. 

Rays  falling  on  any  part  of  a  concave  mirror  parallel  to  its 
axis,  will  all  be  brought  to  a  focus  at  the  same  point,  if  the  cur- 
vature of  the  mirror  be  that  of  a  parabola.  For  then,  according 
to  a  property  of  the  parabola,  all  diameters,  or  lines  parallel  to 
the  axis,  and  a  line  drawn  from  the  focus  to  the  point  where  the 
diameters  meet  the  curve,  make  equal  angles  with  the  tangents 
at  those  points.*  But  these  equal  angles  are  the  complements 
of  the  angles  of  incidence  and  reflexion,  which  are  also  equal. 
Wherefore  rays,  parallel  to  the  axis,  will  be  reflected  into  the 
lines  which  all  meet  at  one  and  the  same  focus. 

725.  DIVERGING  RAYS,  incident  upon  a  concave  mirror,  are  col- 
lected into  a  focus,  which  changes  its  situation  as  the  distance  of  the 
radiant  from  the  mirror  is  changed,  conformably  to  the  law,  that  the 
angle  of  incidence  is  equal  to  the  angle  of  reflexion  made  with  the 
radius  of  concavity.^ 

If  the  radiant  point  be  further  from  the  mirror  than  the  center, 
as  at  A,  (Fig.  245,)  then  the  focus  will  be  between  the  center 
and  the  mirror  ;  if  the  radiant  be  at  the  center,  the  rays  will  re- 
turn to  the  center  again  ;  if  the  radiant  comes  still  nearer  to  the 
mirror,  the  focus  will  pass  to  the  other  side  of  the  center  and 
continue  to  recede  from  it,  until  the  radiant  has  arrived  at  the 
place  of  the  focus  of  parallel  rays,  when  the  focus  on  the  other 
side  of  the  center  will  be  thrown  to  an  infinite  distance  ;  and 
finally,  if  the  radiant  be  brought  nearer  to  the  mirror  thart*  the 
principal  focus,  the  rays  will  go  out  diverging,  and  will  never 
come  to  a  focus  ; — all  which  is  evident  from  the  general  law  of 
reflexion,  the  situation  of  each  reflected  ray  being  easily  deter- 
mined by  that  of  the  incident  ray  with  respect  to  the  perpen- 
dicular, that  is,  the  radius  of  the  mirror.  Thus,  the  rays  emitted 
from  A  will  be  collected  in  a ;  those  from  C  will  return  to  C 
again  ;  those  from  a  will  be  collected  in  A ;  those  from  F,  the 

*  Conic  Sections. 

t  The  several  cases  will  be  the  more  easily  remembered,  by  keeping  in  mind  the 
situation  of  the  incident  ray  relatively  to  the  perpendicular  ;  that  is,  the  radius  of  con- 
cavity. 


focus  of  parallel  rays,  will 
be  reflected  into  the  parallel 
lines  cf,  cf ;  and  those  from 
D  into  the  diverging  lines 
cd,  cd,  which  will  appear  to 
proceed  from  A'.  Again,  if 
the  radiant  is  first  placed 
near  the  mirror,  and  re- 
moved from  it  by  succes- 
sive steps,  just  the  converse 
effects  will  follow.  Hence, 
the  radiant  and  its  corres- 
ponding focus  are  denomi- 
nated conjugate  foci.  In  the 
foregoing  experiment,  the 
conjugate  foci  approach  one 
another — meet  in  the  center 
of  concavity — pass  to  differ- 
ent sides  of  that  center — and 
afterward  recede  from  one 
another,  until  the  focus  near- 
est to  the  mirror  arrives  at 
the  focus  of  parallel  rays, 
when  the  two  conjugate 
foci  are  separated  to  the 
greatest  possible  distance 
from  each  other. 


726.  Parallel  rays  incident  upon  a  CONVEX  MIRROR,  are  made  lu 
diverge  as  from  a  point  behind  the  mirror. 

Let  MN  (Fig.  246)  be  a  convex  mirror  whose  center  is  C,  and 
let  AM,  AD,  AN,  be  parallel  rays  falling  upon  it.  Continue  the 


Fig.  246. 


lines  CM  and  CN  to  E,  E,  and  ME, 
NE,  will  be  perpendicular  to  the 
surface  of  the  mirror  at  the  points 
M  and  N.  The  rays  AM,  AN,  will 
therefore  be  reflected  in  the  direc- 
tions MB,  NB,  the  angles  of  reflex- 
ion  EMB,  ENB,  being  equal  to  the 
angles  of  incidence  EMA,  ENA. 
By  continuing  the  reflected  rays 
BM,  BN  backward,  they  will  be 
found  to  meet  behind  the  mirror  at 
F,  their  virtual  focus. 


727.  Diverging  rays  incident  upon  a  convex  mirror,  are  made 
to  diverge  as  from  a  point  behind  the  mirror,  and  nearer  to  it  than 
the  virtual  focus  of  parallel  rays. 

62 


490 


NATURAL    PHILOSOPHY. 


Let  MN  (Fig.  247,)  be  a 
convex  mirror,  C  its  center 
of  convexity,  and  AM,  AN, 
rays  diverging  from  A, 
which  fall  upon  the  mir- 
ror at  the  points  M,  N. 
The  lines  CME  and  CNE,C 
will  be,  as  before,  perpen- 
dicular to  the  mirror  at  M 
and  N ;  and,  consequently, 
if  we  make  the  angles  of 
reflexion  EMB,  ENB,  equal 
to  the  angles  of  incidence  EMA,  ENA,  then  MB,  NB,  will  be 
the  reflected  rays,  which,  when  continued  backward,  will  meet 
at  F,  their  virtual  focus  behind  the  mirror.  By  comparing  Figs. 
246  and  247,  it  will  be  obvious  that  the  ray  AM  in  Fig  247,  is 
further  from  ME  than  in  Fig.  246,  and  consequently,  the  reflected 
ray  MJ3  must  also  be  further  from  it.  Hence,  as  the  same  is 
true  of  the  ray  NB,  the  point  F,  where  these  rays  meet,  must  be 
aearer  D  in  Fig.  247,  than  in  Fig.  246  ;  that  is,  in  the  reflexion 
of  diverging  rays,  the  virtual  focal  distance  DF,  is  less  than  for 
parallel  rays.  For  the  same  reason,  if  »we  suppose  the  radiant 
point  A  to  approach  the  mirror,  the  virtual  focus  F  will  ap- 
proach it ;  and  when  A  arrives  at  D,  F  will  also  arrive  at  D. 
In  like  manner,  if  A  recedes  from  the  mirror,  F  will  recede  from 
it ;  and  when  A  is  infinitely  distant,  or  when  the  rays  become 
parallel,  as  in  Fig.  246,  F  will  reach  the  place  of  the  virtual 
focus  of  parallel  rays. 


CHAPTER  III. 

\ 

OF  IMAGES  FORMED  BY  PLANE,  CONCAVE,  AND  CONVEX  MIR- 
RORS. 

728.  In  all  mirrors,  plane  or  spherical,  the  place  of  the  imagi- 
nary radiant,  is  the  intersection  of  the  perpendicular  from  the  radi- 
ant point  of  the  object  to  the  mirror,  with  any  reflected  ray. 

All  the  rays  which  diverge  from  any  point  in  the  object  before 
reflexion,  appear,  after  reflexion,  to  diverge  from  one  and  the 
same  point,  namely,  from  the  imaginary  radiant.  (See  Fig.  243.) 
And  since  the  perpendicular  ray  RP  is  reflected  back  to  R,  the 
imaginary  radiant  must  be  in  that  line  produced ;  and  eince  the 
imaginary  radiant  must  likewise  be  in  any  other  reflected  ray  as 
AD,  produced,  it  must  be  in  the  intersection  of  the  two  lines. 


Fig.  248. 


The  same  reasoning  would  apply  to  a  concave  or  a  convex  sur- 
face, since  the  reflexion  at  any  point  of  such  a  surface,  is  the 
same  as  it  would  be  from  a  plane  surface  which  is  a  tangent  to 
the  curve  at  that  point. 

729.  When  any  object  is  placed  before  a  PLANE  mirror,  the  im- 
age of  it  appears  at  the  same  Distance  behind  it,  of  the  same  mag- 
nitude, and  equally  inclined  to  it. 

Let  MN  (Fig.  248.)  be  a  plane  mirror, 
and  AB  an  object  placed  before  it,  and 
let  the  position  of  the  object  be  such  that 
the  reflected  rays  may  enter  an  eye  plac- 
ed at  H.  From  A  and  B  let  fall  upon 
the  mirror  the  perpendiculars  Aa,  Bb. 
Then  the  rays  AF,  AG,  diverging  from 
A,  will  be  reflected  in  the  lines  FH,  GK, 
as  if  they  came  from  the  point  a,  so  situ-  M 
ated  that  EA=Ea,  (Art.  723  ;)  and  hence 
the  point  A  will  be  seen  at  a,  as  far  be- 
hind the  mirror  as  A  is  before  it.  In  like 
manner,  it  may  be  shown  that  the  point. 
B  of  the  object,  will  appear  at  b,  so  situ- 
ated that  GB=G6.  By  taking  any  other  rays  at  pleasure,  divergent 
from  any  other  point  of  the  object  AB,  it  may,  in  a  similar  man- 
ner, be  shown,  that  they  will  have  their  foci  in  points  of  the 
line  ab,  formed  by  drawing  perpendiculars  from  the  given 
points  of  the  object.  Now,  since  GB=Gb  and  aG&=BGK=AGB, 
Ga=GA,  .'.  AB=a&.  That  is,  the  magnitude  of  the  image 
equals  that  of  the  object.  From  A  and  a  draw  the  perpendicu- 
lars AC,  ac  ;  then  the  angle  BAC=bac,  that  is,  the  object  and 
the  image  are  equally  inclined  to  the  mirror. 

Hence  objects  that  are  perpendicular  to  the  horizon,  seen  in  a 
plane  mirror,  appear  inverted,  the  highest  point  of  the  object  be- 
ing the  lowest  point  of  the  image. 

If  a  plane  mirror  is  inclined  to  the  horizon  at  an  angle  of  45°, 
an  object  parallel  to  the  horizon  will  appear  erect,  and  perpen- 
dicular objects  will  appear  horizontal.  For,  since  the  image  has 
the  same  inclination  to  the  mirror  as  the  object  has,  when  the 
angle  made  by  the  object  is  45°,  that  made  by  the  image  must 
be  45°  also,  and  both  together  must  make  90°. 

730.  If  the  image  of  an   object   is  seen  by  reflexion,  from 
two  plane  mirrors,  the  reflexion  being  in  a  plane  perpendicular 
to  their  common  section,  the  angular  deviation  of  the  image  from 
the  object,  will  be  equal  to  twice  the  inclination  of  the  reflecting 
mirrors. 

Let  AB,  CD,  (Fig.  249,)  be  two  plane  mirrors,  inclined  to  each 
other  in  any  angle.  Produce  AB  and  CD  until  they  meet  in  G  ; 


492  NATURAL   PHILOSOPHY. 

then  AGO  will  be  the  angle  of  in- 
clination of  the  mirrors.  Let  SB  be 
a  ray  of  light  proceeding  from  any 
distant  object,  as  a  star,  and  reflected 
from  AB  to  CD,  and  from  CD  to  the 
eye  at  H  ;  then  the  image  of  S  will 
be  at  O  in  the  line  HD  produced^ 
Also  produce  SB  to  H,  then  SHO 
will  be  the  angular  deviation  of 
the  image  from  the  object,  the  line 
of  common  section  of  the  mirrors 
passing  through  G,  perpendicular 
to  the  plane  AGC.  It  is  required 
to  prove  that  the  angle  SHO  is 
double  the  angle  AGC.  Because 
HBG=ABS=GBD  /.HBD=2GBD. 
In  like  manner  BDO=2BDC. 

But,  SHO=BDO-HBD--=2BDC 
-2GBD=2BGD. 

Therefore,  SHO-2BGD. 
,  Hence,  when  a  plane  mirror  revolves  on  an  axis,  the  angular 
velocity  of  the  reflected  ray  is  double  that  of  the  mirror.  There- 
fore, by  turning  a  mirror  through  45°,  the  image  is  carried  through 
90°,  so  that  a  mirror  set  at  an  angle  of  45°  with  the  horizon,  rep- 
resents horizontal  objects  in  a  perpendicular  position,  and  perpen- 
dicular objects  on  a  horizontal  level,  agreeably  to  the  last  article 

Upon  the  foregoing  proposition  depends  a  principle  employed 
in  Hadley's  Quadrant,  in  which  two  mirrors,  inclined  to  one  an- 
other, measure  the  angular  distance  between  two  objects,  by 
indicating  the  arc  through  which  the  image  of  one  of  them 
must  be  made  to  pass,  in  order  to  carry  it  over  that  distance.* 
Thus,  if  in  order  to  make  the  image  of  a  star  descend  to  the 
horizon,  the  mirror  that  reflects  it  must  be  turned  20°,  the  alti- 
tude of  the  star  is  40°.  Hence,  an  octant  only  is  required  to 
measure  a  quadrant,  or  an  angle  of  90°. 

731.  When  the  object  is  parallel  to  a  plane  mirror,  the  length  or 
breadth  of  that  part  of  the  mirror  upon  which  the  image  appears,  is 
to  the  length  or  breadth  of  the  object,  as  any  rejlected  ray  is  to  the 
sum  of  the  incident  and  rejlected  rays. 

If  the  object  DE  (Fig.  250,)  is  parallel  to  the  mirror  AB,  and 
the  image  LM  is  seen  by  the  eye  at  C,  then  FN,  the  length 
of  that  part  of  the  mirror  which  is  taken  up  by  the  image, 
subtends  the  angle  LCM,  under  which  the  image  appears. 
Now  the  length  of  the  image  LM  is  equal  to  the  length  of 
the  object  DE.  (Art.  729.)  And  because  FN  is  parallel  to  LM, 

*  See  Day's  Navigation  and  Surveying,  Art.  91. 


Fig.  250. 


OPTICS.  493 

.-.FN:LM::CF:CL.  ButCFis 
the  reflected  ray,  and  CL  is  equal 
to  the  sum  of  the  incident  and  the 
reflected  rays. 

Hence,  an  object  which  is  not 
wholly  visible  when  the  eye  is  at 
a  certain  distance  from  the  mirror, 
may  become  so  by  bringing  the 
eye  nearer  to  the  mirror ;  for  in 
proportion  as  the  ratio  of  the  re- 
flected ray  to  the  sum  of  the  incident  and  reflected  rays  is  di- 
minished, in  the  same  proportion  is  the  part  of  the  mirror  required 
to  comprehend  the  entire  image  diminished. 

If  a  spectator  sees  himself  entirely  in  a  plane  mirror  placed 
parallel  to  him,  the  mirror  must  be  half  as  long  as  himself.  For 
then  the  incident  and  reflected  rays  will  be  equal,  and  conse- 
quently the  latter  equal  to  half  the  sum  of  the  two,  and  hence 
the  mirror  must  be  half  the  length  of  the  object. 

732.  When  an  object  is  placed  between  two  parallel  plane  reflect- 
ors, a  row  of  images  is  formed  in  each  mirror,  appearing  in  a 
straight  line  behind  each  to  an  indefinite  extent. 

Let  there  be  two  plane  reflectors,  parallel  to  each  other ;  and 
let  an  object,  a  candle  for  example,  be  placed  between  them. 
An  image  of  the  candle  will  be  formed  in  each  mirror,  as  far 
behind  it  as  the  object  is  before  it.  Again,  each  of  these  im- 
ages becomes  in  its  turn  a  new  object  to  the  opposite  mirror, 

Fig.  251. 
A  C 


and  forms  a  corresponding  image  as  far  behind  that  mirror  as  it 
is  itself  before  it,  and  thus  the  images  are  repeated  in  a  right 
line  until  the  light  becomes  too  feeble  to  be  visible.  Thus  let 
AB,  CD,  (Fig.  251,)  be  two  plane  mirrors,  and  E  an  object  be- 
tween them  ;  two  images  will  be  formed  of  E  at  E'  and  E',  from 
which  virtual  foci  the  light  will  emanate,  and  will  strike  upon  the 
opposite  mirrors,  respectively,  in  the  same  manner  as  if  it  came 
from  luminous  bodies  placed  at  those  points  ;  hence  two  images 


494  NATURAL   PHILOSOPHY. 

of  E'  and  E'  will  be  formed  at  E"  and  E"  ;  and  thus  a  succes- 
sion of  images  will  arise  to  an  indefinite  extent ;  but  since  a 
part  of  the  light  is  lost  at  every  reflexion,  each  succeeding  image 
is  fainter  than  the  preceding.  The  Endless  Gallery  is  formed  on 
this  principle.  It  consists  of  a  box,  in  the  opposite  sides  of 
which  are  placed  two  parallel  reflectors,  and  between  them  a 
number  of  objects  are  placed,  which  are  repeated  in  an  endless 
succession. 

733.  If  an  object  be  placed  between  two  plane  reflectors  inclined 
to  each  other,  the  images  formed  will  lie  in  the  circumference  of  a 
circle,  whose  center  is  in  the  intersection  of  the  two  planes,  and 
whose  radius  is  the  distance  of  the  object  from  that  point. 

Let  AB,  AC,  (Fig.  252,)  be  Fig.  252. 

two  plane  reflectors  inclined 
at  the  angle  BAG,  and  E  an 
object  placed  between  them. 
Draw  EF  perpendicular  to 
AB,  and  produce  it  to  G, 
making  FG=EF ;  then  the 
rays  which  diverge  from  E 
and  fall  upon  AB  will,  after 
reflexion,  diverge  from  G  ;  or 
G  will  be  an  image  of  E.* 
From  G,  draw  GH  perpendic- 
ular to  AC,  and  produce  it  to 
I,  making  HI=GH,  and  I  will 
be  a  second  image  of  E,  &c. 
Again,  draw  ELM  perpen- 
dicular to  AC,  and  make  LM=EL  ;  also  draw  MNO  perpendic- 
ular to  AB,  and  make  NO=MN,  &c.  Therefore,  the  successive 
images  formed,  beginning  on  the  side  of  AB,  are  G,  I,  K,  V ;  and 
those  on  the  side  of  AC,  are  M,f),  P,  Q.  Then,  since  EF  is  equal 
to  FG,  and  AF  common  to  the  triangles  AFG,  AFE,  and  the  an- 
gles at  F  are  right  angles,  AG  is  equal  to  AE.  In  the  same 
manner  it  may  be  shown  that  AM,  AO,  AI,  &c.,  are  severally 
equal  to  AE  ;  and  of  course,  the  points  G,  M,  O,  I,  &c.  are  in  the 
circumference  of  a  circle  whose  center  is  A  and  radius  AE.  ^ 

If  the  angle  BAG  is  finite,  the  number  of  images  will  be  limited. 
For  BA  and  CA  being  produced  to  R  and  S,  the  rays  which  are 
reflected  from  either  surface,  diverging  from  any  point  Q,  and 
between  S  and  R,  will  not  meet  the  other  reflector,  since  it  is 
not  before  either  reflector,  but  behind  both. 

*  The  eye  is  supposed  to  be  situated  between  the  planes  of  the  mirrors.  The  learn- 
er may  find  some  difficulty  in  conceiving  how  rays  of  light  can  proceed  from  G,  a 
point  behind  the  mirror  ;  but,  it  must  be  recollected  that  this  is  only  an  imaginary 
point,  from  which  the  rays  that  are  reflected  from  the  mirror  into  the  line  GH  seem 
to  emanate. 


OPTICS. 


495 


If  we  consider  the  whole  angular  opening  of  the  mirrors, 
namely,  the  sector  ABC,  as  the  object,  images  of  it  will  be  form- 
ed in  a  circle  as  of  any  other  object. 

If  the  inclination  of  the  mirrors  gradually  diminishes,  the  mag- 
nitude of  the  sectors  will  also  be  diminished,  and  the  number  of 
repetitions  of  them  increased  in  the  same  proportion.  The 
number  of  images  of  any  object  placed  between  the  mirrors, 
will  be  in  like  manner  increased  as  the  inclination  of  the  mir- 
rors is  diminished ;  and  since,  when  the  angle  of  inclination  is 
very  small,  the  mirrors  approach  the  situation  of  parallel  mir- 
rors, so  the  number  of  images  approach  to  infinity. 
* 

734.  As  the  learner  sometimes  experiences  a  difficulty,  in  con- 
ceiving clearly  the  course  which  the  rays  take  in  forming  the 
successive  images,  we  subjoin  a  brief  illustration.  It  may  be 
premised,  that  whatever  turns  a  ray  of  light  may  take,  in  pass- 
ing from  an  object  to  the  eye,  the  object  will  be  seen  in  that  di- 
rection in  which  the  light  finally  meets  the  eye.  Thus  in  forming 
some  of  the  foregoing  linages,  a  ray  of  light  undergoes  three  or 
four  reflexions,  in  different  directions,  but  still  the  last  ray  that 
meets  the  eye  will  fix  the  position  of  the  image. 


Fig.  253. 


Let  RS,  RT,  (Fig.  253,) 
be  two  plane  mirrors  in- 
clined to  each  other,  and 
Q  a  luminous  object  be- 
tween them.  Let  the  eye 
be  situated  at  any  conve- 
nient point, as  O,  and  let  A, 
B,  C,  D,  be  the  several  im- 
ages. Draw  a  line  from 
O  to  any  one  of  these  im- 
ages, as  D,  and  from  the 
point  where  this  line  inter- 
sects the  speculum,  draw  a 
line  to  the  next  preceding 
image,  and  from  the  point 
where  this  line  meets  the 
speculum,  draw  a  line  to 
the  next  preceding  image, 
and  so  on  back  to  the  object  Q. 

Now  the  image  A  is  seen  by  the  ray  Qot,  aO  ;  B,  by  Q6,  be, 
cO  ;  C,  by  Qrf,  de,  ef,fO  ;  and  D,  by  Qg,  gh,  hi,  ik,  AO. 

For,  since  S«Q=SaA=RaO  .-.SaQ=R«O,  and  therefore  Qa 
being  the  incident,  «O  must  be  the  reflected  ray  ;  and  the  same 
may  be  proved  in  each  of  the  several  cases  of  reflexion. 

735.  It  is  found  by  experiment,  that  when  a  pencil  of  light  is 
incident  perpendicularly  upon  water,  only  18  rays  out  of  1000 


496  NATURAL   PHILOSOPHY. 

are  reflected,  while  the  greater  part  of  the  remaining  rays  are 
transmitted.  As  the  angle  of  inclination  is  increased,  the  propor- 
tion of  rays  reflected  is  also  rapidly  increased,  till  at  an  angle  of 
75°  the  reflexion  is  211  rays  ;  at  85°,  501  ;  and  at  89°,  692.  In 
glass,  25  out  of  1000  are  reflected  at  a  perpendicular  incidence  ; 
and  the  glass  always  reflects  more  light  than  water,  till  we  reach 
very  great  angles  of  incidence,  such  as  87^°,  when  it  reflects 
only  584  rays,  while  water  reflects  614. 

736.  The  image  is  inverted,  when  the  rays  of  light  which  come 
from  one  extremity  of  the  object,  cross  those  which  come  from  the 
other  extremity,  before  they  meet  in  the  corresponding  points  of  the 
image. 

Thus,  let  AB  (Fig.  254,)  repre- 
sent a  board  or  screen  having  a 
small  opening  through  which  rays 
of  light  may  pass.  Since  light 
passes  in  right  lines,  it  is  obvious 
that  rays  proceeding  from  C,  the 
highest  point  in  the  object  CD, 
will  pass  to  C',  while  rays  from 
D,  the  bottom  of  the  object,  will 
pass  to  D',  crossing  each  other  at  the  orifice,  and  thus  forming 
an  inverted  image  of  the  object,  D'C'.  In  like  manner  rays 
coming  from  all  the  intermediate  points,  on  opposite  sides,  will 
cross  before  they  take  their  respective  places  in  the  image. 

737.  When  an  object  is  placed  before  a  CONCAVE  mirror,  the  im- 
age of  it  has  various  magnitudes  and  positions,  depending  on-  the 
distance  of  the  object  from  the  mirror.* 

1.  When  an  object  is  between  the  mirror  and  the  focus  of  par- 
allel rays,  the  image  appears  behind  the  mirror,  and  is  more  dis- 
tant from  it,  and  larger  than  the  object.  Let  MN  (Fig.  255,)  be 
the  concave  mirror,  F  its  principal  focus,  C  its  center,  and  AB 
the  object.  From  C  draw  CAa,  CBfc,  passing  through  A  and  B, 
and  let  the  object  be  so  placed  that  the  reflected  rays  will  reach 
the  eye  at  H.  The  rays  AD,  AG,  proceeding  from  A,  will  be  re- 
flected to  the  eye  at  H,  making  equal  angles  with  the  perpen- 
dicular CD,  and  they  will  diverge  as  if  they  had  come  from  a 
remote  point  a,  situated  in  the  intersection  of  those  rays  with 
the  perpendicular  CAa,  (Art.  723.)  In  like  manner,  the  rays  Ed, 
Eg,  will  enter  the  eye  at  H,  as  if  they  had  proceeded  from  b,  a 
point  where  they  intersect  CB&.  Since  the  rays  diverge  less 
than  before  reflexion,  (Art.  725,)  these  points,  a,  b,  will  be  further 

*  These  different  places  of  the  image  depend  on  the  principles  demonstrated  in 
Art.  725,  and  they  will  be  easily  remembered  by  considering  the  relation  of  the  inci- 
dent rays  to  the  perpendicular,  that  is,  the  radius  of  tbs  concavity,  conformably  to 
the  general  laws  of  reflexion. 


497 


from  MN  than  A  and  B  are,  and  from  similar  triangles,  the  im- 
age ab  will  be  greater  than  AB,  in  the  ratio  of  Cb  to  CB. 
Fig.  255.  Fig.  256. 


a.  When  the  object  is  placed  in  the  principal  focus,  the  raya, 
will  go  out  parallel,  and  will  never  come  together  so  as  to  form 
an  image  of  themselves,  nor  will  they  proceed  from  any  point 
behind  the  mirror,  so  as  to  form  an  imaginary  image,  like  that 
of  case  1. 

3.  When  the  object  is  situated  between  the  principal  focus  and 
the  center,  the  image  is  formed  on  the  other  side  of  the  center, 
and  is  inverted  and  larger  than  the  object.     Let  MN,  (Fig.  256,) 
be   the   mirror,  C  its  center,  F  its  focus,  and  AB  the  object. 
Through  C  draw  the  lines  CA,  CB,  and  continue   them   back- 
ward to  a  and  b.     Then  let  AD,  AG,  and  Ed,  B^,  be  two  sets  of 
rays  flowing  from  the  extremities  A,  B.     These  rays  will,  after 
reflexion  in  the  directions  Da,  Ga,  and  db,  gb,  meet  the  perpen- 
dicular lines  C«,  Cb,  in  the  points  a,  b,  at  a  greater  distance  from 
the  mirror  than  the  center  C,  (being  -reflected  to  the  other  side 
of  the  radius,)  and  will  there  form  an  image  of  those  points  of 
the  object.     (Art.  725.)     The  image  is  therefore  more  remote 
from  C  than  the  object  is,  and  the  size  of  the  one  will  be  to  that 
of  the  other  as  aC  is  to  AC  ;  that  is,  the  image  will  be  larger  than 
the  object.* 

4.  When  the  object  is  situated  beyond  the  center,  the  image 
will  then  be  formed  between  the  center  and  the,  principal  focus, 
and  will  be  inverted  and  less  than  the  object.     This  is  the  con- 
verse of  the  preceding,  and  will  be  made  obvious  by  considering 
the  rays  as  first  flowing  from  ab  and  converging  to  AB.     When 
the  middle  part  of  the  object  is  placed  in  the  center  of  the  mirror, 
the  object  will  coincide  with  the  image,  and  the  image  will  be 
inverted.     That  the  center  of  the  image  will  coincide  with  that 
of  the  object,  may  be  inferred  from  Art.  725  ;  the  reflected  ray 
being  returned  back  in  the  incident  ray  or  perpendicular  ;  and 

*  For,  since  the  rays  AD,  Da,  make  equal  angles  with  the  radius  CD,  C«  is  greater 
than  CA,  and  consequently,  from  similar  triangles,  a6>AB.  See  Fig.  245. 

63 


498 


NATURAL   PHILOSOPHY. 


rays  proceeding  from  the  extremities  of  the  object  A  and  B,  will 
make  equal  angles  with  this  perpendicular  on  the  different  sides  of 
it,  and  therefore  an  inverted  image  will  fall  back  upon  the  object. 

738.  The  following  experiments,  which  may  be  easily  repeat- 
ed, will  serve  to  render  familiar  the  different  cases  above  demon- 
strated. 

We  will  suppose  a  lighted  candle  to  be  placed  very  near  to  a 
concave  mirror :  it  will  form  no  image  before  it,  because  the 
rays  go  out  still  diverging,  but  we  see  an  enlarged  image  of  the 
candle  behind  the  mirror.  As  the  radiant  is  withdrawn  from  the 
mirror  toward  the  principal  focus,  the  image  will  rapidly  recede  on 
the  other  side  of  the  mirror,  and  grow  larger  and  larger  until  the  ra- 
diant reaches  the  focus,  when  the  image  will  suddenly  disappear. 
On  removing  the  radiant  a  little  further, the  image  will  be  found  at 
a  distance  before  the  mirror,  and  very  much  enlarged.  As  the 
radiant  approaches  the  center,  the  image  approaches  it  rapidly 
on  the  other  side  of  it,  constantly  diminishing  in  size,  until  they 
both  meet  and  coincide  in  the  center.  Removing  the  radiant 
still  further,  the  image  appears  again  between  the  center  and 
the  focus,  diminished  in  size,  and  slowly  approaching  the  focus 
as  the  radiant  recedes,  but  never  reaches  it,  unless  when  the  ra- 
diant may  be  considered  as  at  an  infinite  distance,  as  in  the  case 
of  the  heavenly  bodies.  By  applying  principles  already  ex- 
plained, the  learner  will  readily  account  for  these  various  ap- 
pearances. 

739.  When  any  object  is  placed  before  a  CONVEX  mirror,  the 
image  of  it  appears  nearer  to  the  surface  of  the  mirror  than  the 
object  is,  and  of  a  less  size. 

Let  MN  (Fig.  257)  be  a  convex  mirror  Fig.  257. 

whose  center  is  C,  and  AB  the  object ; 
and  let  the  position  of  the  object  be  such 
that  a  reflected  ray  may  enter  the  eye 
placed  at  H.  From  C  draw  CA,  CB, 
cutting  the  mirror  MN  in  E  and  F.  The 
rays  AF,  AG,  will  be  reflected  to  H  and 
K,  making -equal  angles  with  the  perpen- 
dicular passing  from  C  through  F  and  G, 
and  will  therefore  enter  the  eye  as  if  they 
came  from  some  point  as  a,  at  the  inter- 
section of  these  rays  with  the  perpendic- 
ular AC ;  consequently,  the  point  A  of 
the  object  will  have  its  image  visible  at  a.  In  like  manner,  rays 
Ef,  Bg-,  falling  upon  the  points  /,  g,  will  be  reflected  to  the  eye 
as  if  they  came  from  b,  the  point  where  they  intersect  the  per- 
pendicular drawn  from  B  to  C.  Now,  as  the  reflected  rays  di- 
verge more  than  the  incident  ones,  the  point  a  will  be  nearer  the 


mirror  than  A,  and  the  image  ab  will  be  less  than  the  object  AB, 
in  the  ratio  of  Cb  to  CB. 

• 

740.  In  spherical  mirrors,  concave  or  convex,  the  diameter  of  the 
object  is  to  the  diameter  of  the  image,  as  the  distance  of  the  object 
from  the  center  is  to  the  distance  of  the  image  from  the  center ;  and 
also  as  the  distance  of  the  object  from  the  surface  is  to  the  distance 
of  the  image  from  the  surface. 

It  is  evident,  from  Figs.  256, 257,  that  the  object  and  the  image 
subtend  each  the  same  angle,  the  former  at  the  center  and  the  lat- 
ter at  the  surface  ;  and  as  they  are  parallel  to  one  another,  their 
lengths  are  as  their  distances  from  these  points  respectively.* 

741.  One  who  looks  into  a  concave  mirror,  sees  his  own  face 
varied  in  the  following  manner. 

When  he  holds  the  reflector  near  to  his  face,  he  sees  his  image 
distinct,  because  the  rays  come  to  the  eye  diverging,  (which  is 
their  natural  state  with  respect  to  near  objects,)  and  enlarged, 
because,  as  the  rays  diverge  less  than  before,  the  image  is  thrown 
back  to  a  greater  distance  behind  the  mirror  than  the  object  is 
before  it,  and  the  magnitude  is  as  that  distance,  by  Art.  740. 
As  he  withdraws  the  eye,  the  image  grows  larger  and  larger, 
until  the  eye  reaches  the  focus.  From  the  focus  to  the  center, 
no  distinct  image  is  seen,  because  the  rays  come  to  the  eye  con- 
verging, a  condition  incompatible  with  distinct  vision.  At  the 
center,  the  eye  sees  only  its  own  image,  since  the  image  is  re- 
flected back  to  the  object  and  coincides  with  it.  Beyond  the 
center,  his  face  will  be  seen  on  the  other  side  of  the  center  be- 
fore the  mirror,  (though  habit  may  lead  him  to  refer  it  to  a  point 
behind  it ;)  and  it  will  be  diminished,  being  nearer  to  the  mirror 
than  the  object  is,  (Art.  740,)  and  inverted,  because  the  pencils 
of  rays  from  the  extreme  points  of  the  object,  cross  each  other 
in  the  focus,  f 

742.  By  the  reflexion  of  light  from  concave  mirrors,  there  are 
exhibited  curves  of  a  peculiar  kind,  called  CAUSTICS  BY  REFLEXION.  J 

Let  MBA  (Fig.  258)  be  a  concave  spherical  mirror  whose  cen- 
ter .is  C,  and  whose  focus  for  parallel  and  central  rays  is  F.  Let 
RMB  be  a  pencil  of  light  falling  on  the  upper  half,  MB,  of  the 
mirror,  at  the  points  1,  2,  3, 4,  &c.  If  we  draw  radii  to  all  these 
points  from  the  center  C,  and  make  the  angles  of  reflexion  equal 
to  the  angles  of  incidence,  we  shall  obtain  the  directions  and  foci 
of  all  the  incident  rays.  The  ray  Rl,  near  the  axis  RB,  will 

*  Euc.  VI,  4. 

t  These  phenomena  may  be  all  observed  with  an  ordinary  concave  shaving  glass. 

t  Called  caustics,  or  burning  points,  because,  since  the  rays  of  light  or  heat  cross 
each  other  in  the  points  that  make  up  these  curves,  the  intensity  of  light  or  heat  is 
twice  as  great  there  as  elsewhere. 


500 


NA1URAL    PHILOSOPHY. 


have  its  conjugate  focus  at/,  between  F  and  the  center  C.     The 
next  ray,  R2,  will  cut  the  axis  nearer  F,  and  so  on  with  all  the 
rest,  the  foci  advancing  from  €  to  F.     By  drawing  all  the  re- 
Fig.  258. 


N    ft 


fleeted  rays  to  these  foci,  they  will  be  found  to  intersect  one  an- 
other, as  in  the  figure,  and  to  form  by  the  intersection  the  caustic 
curve  M/.  They  are  so  called  because,  in  consequence  of  the  in- 
tersections of  the  rays  in  the  points  forming  these  curves,  those 
points  are  brighter,  or,  where  heat  is  reflected,  hotter,  than  the  con- 
tiguous spaces.  If  the  light  had  also  been  incident  on  the  lower 
half  of  the  mirror,  a  similar  caustic,  shown  by  a  dotted  line,  would 
also  have  been  formed  between  N  and  f.  If  we  suppose,  there- 
fore, the  point  of  incidence  to  move  from  M  to  B,  the  conjugate 
focus  of  any  two  contiguous  rays,  or  an  infinitely  slender  pencil 
diverging  from  R,  will  move  along  the  caustic  from  M  to/.* 

743.  Concave  mirrors,  in  consequence  of  the  property  they 
have  of  forming  images  in  the  air,  were,  in  a  less  enlightened 
age  than  the  present,  frequently  employed  by  showmen  for  ex- 
hibiting surprising  appearances.  The  mirror  was  usually  con- 
cealed behind  a  wall,  and  the  object,  which  might  be  a  skull,  a 
dagger,  &c.,  was  placed  between  it  and  the  wall,  and  strongly 
illuminated.  The  rays  proceeding  from  the  object,  fell  upon  the 
mirror,  and  were  reflected  by  it  through  an  opening  in  the  wall, 
and  brought  to  a  focus,  so  as  to  form  an  image  in  the  same  room 
with  the  spectator.  If  a  fine  transparent  cloud  of  blue  smoke  is 
raised,  by  means  of  a  chafing  dish,  around  the  focus  of  a  large 
concave  mirror,  the  image  of  any  highly  illuminated  object  will 
be  depicted  in  the  middle  of  it  with  great  beauty.  A  dish  of 


*  These  curves  may  be  seen  on  the  surface  of  milk,  pJsced  in  a  white  bowl  or  tea- 
cup,  set  in  the  sun. 


OPTICS.  501 

fruit  thus  represented  invites  the  spectator  to  taste,  but  the  in- 
stant he  reaches  out  his  hand  a  drawn  dagger  presents  itself.* 

744.  Concave  mirrors  have  been  used  as  lighthouse  reflectors, 
and  as  burning  instruments.  When  used  in  lighthouses,  they  are 
formed  of  copper  plated  with  silver,  and  they  are  hammered  into 
a  parabolic  form,  and  then  polished  with  the  hand.  A  lamp 
placed  in  the  focus  of  the  parabola,  will  have  its  divergent  light 
thrown,  after  reflection,  into  something  like  a  parallel  beam, 
which  will  retain  its  intensity  to  a  great  distance. 

When  concave  mirrors  are  used  for  burning,  they  are  general- 
ly made  spherical,  and  regularly  ground  and  polished  upon  a 
tool,  like  the  specula  used  in  telescopes.  The  most  celebrated  of 
these  were  made  by  M.  Villele,  of  Lyons,  who  executed  five 
large  ones.  One  of  the  best  of  them,  which  consisted  of  copper 
and  tin,  was  very  nearly  four  feet  in  diameter,  and  its  focal 
length  thirty-eight  inches.  It  melted  the  metals,  as  silver  and 
copper,  and  even  some  of  the  more  infusible  earths. 

Burning  mirrors,  however,  have  sometimes  been  constructed 
on  a  much  larger  scale,  by  combining  a  great  number  of  plane 
mirrors.  It  is  supposed  that  it  was  a  mirror  of  this  kind  which 
Archimedes  employed  in  setting  fire  to  the  Roman  fleet  under 
Marcellus.  Athanasius  Kircher,  who  first  proved  the  efficacy  of 
a  union  of  plane  mirrors,  went  with  his  pupil  Scheiner  to  Syra- 
cuse, to  examine  the  position  of  the  hostile  fleet ;  and  they  were 
both  satisfied  that  the  ships  of  Marcellus  could  not  have  been 
more  than  thirty  paces  distant  from  Archimedes. 

BufFon,  the  celebrated  naturalist,  constructed  a  burning  ap- 
paratus upon  this  principle,  which  may  be  easily  explained.  He 
combined  one  hundred  and  sixty-eight  pieces  of  mirror,  six  inches 
by  eight,  so  that  he  could,  by  a  little  mechanism  connected  with 
each,  cause  them  to  reflect  the  light  of  the  sun  upon  one  spot. 
Those  pieces  of  glass  were  selected  which  gave  the  smallest 
image  of  the  sun  at  two  hundred  and  fifty  feet.  With  one  hun- 
dred and  fifty-four  mirrors,  he  was  able  to  fire  combustibles  at 
the  distance  of  two  hundred  and  fifty  feet. 


CHAPTER  IV. 

OF  THE  REFRACTION  OF  LIGHT. 

745.    WHEN  the  rays  of  light  pass  out  of  one  medium  into 
another,  as  out  of  air  into  water,  they  are  bent  out  of  their  pre- 

vious  direction  ;  and  hence, 

*  Button's  Recreations. 


502  NATURAL   PHILOSOPHY. 

Refraction  is  the  change  of  direction  which  light  undergoes  by 
passing  out  of  one  medium  into  another. 

The  lines  which  a  ray  describes  before  and  after  refraction  are 
called  incident  and  refracted  rays ;  the  angle  contained  between 
the  incident  ray  and  a  perpendicular  to  the  surface  drawn  from 
the  point  on  which  the  ray  falls,  is  called  the  angle  of  incidence ; 
the  angle  contained  between  the  refracted  ray  and  the  perpen 
dicular,  is  called  the  angle  of  refraction.  The  angle  which  the 
refracted  ray  makes  with  its  previous  line  of  direction  is  called 
the  angle  of  deviation.  These  several  definitions  the  learner  will 
easily  apply  to  the  following  figure.  Thus  AC,  (Fig.  259,)  is  the 
incident,  and  CE  the  refracted  Fig.  359. 

ray ;  ACD  is  the  angle  of  inci- 
dence, EOF  the  angle  of  re- 
fraction, GCE  the  angle  of  de- 
viation. It  is  a  general  fact,  to 
which  there  are  only  a  few  ex- 
ceptions, that  a  ray  of  light  in 
passing  out  of  a  rarer  into  a 
denser  medium  is  refracted  to- 
ward the  perpendicular  to  the 
surface  ;  and  in  passing  out  of  a 
denser  into  a  rarer  medium,  it 
isrefracted/rowi  the  perpendic- 
ular. But  the  chemical  consti- 
tution of  bodies,  as  well  as 
their  density,  sometimes  affects 
their  refracting  power.  Thus,  inflammable  bodies,  as  sulphur, 
amber,  and  essential  oils,  have  a  very  great  refracting  power  in 
comparison  with  other  bodies  ;  and  in  a  given  instance,  a  ray  of 
light  in  passing  out  of  one  of  these  substances  into  another  of 
greater  density  but  of  less  refractive  power,  might  not  be  turned 
toward,  but  from,  the  perpendicular. 

746.  When  a  ray  of  light  passes  from  one  medium  into  another 
of  different  density,  the  sines  of  the  angle  of  incidence  and  refrac- 
tion have  always  the  same  ratio  to  each  other  ;  and  the  incident  and 
refracted  rays  are  in  the  same  plane. 

This  proposition  may  be  proved  by  experiment.*  Let  AC 
(Fig.  259,)  be  a  ray  of  light  incident  upon  the  surface  RS  of 
water,  or  any  other  medium.  This  ray,  instead  of  proceeding 
directly  forward  in  AC  produced,  is  bent  or  refracted  at  C  into 
the  direction  CE.  In  like  manner,  another  rayaC,  incident  upon 
the  same  point  C,  is  found  to  be  bent  or  refracted  into  the  line 
Ce.  Through  the  point  C  draw  the  line  PCQ  perpendicular  to 

*  For  a  theoretical  demonstration,  see  Newton's  Optics,  or  Encyc.  Metropolitan!. 


OPTICS.  503 

the  refracting  surface  RS,  and  upon  C  as  a  center,  describe  a 
circle  APQ.  If  we  now  compare  the  angles  of  refraction  with 
the  corresponding  angles  of  incidence,  we  shall  perceive  no  par- 
ticular relation  between  them,  except  that  in  general,  one  in- 
creases or  diminishes  with  the  other ;  but  if  we  compare  the 
sines  of  these  angles,  viz.  AD  with  EF,  and  ad  with  ef,  we  shall 
find  that  the  ratio  of  the  one  to  the  other  is  constant,  so  that  AD 
is  always  to  EF  as  ad  to  ef,  whatever  is  the  value  of  the  angles 
of  incidence  or  refraction.  If  the  Fig.  260. 

surface  RS  is  that  of  water,  into 
which  a  ray  passes  from  the  at- 
mosphere, the  ratio  of  the  sines 
of  incidence  and  refraction  will 
be  as  4  to  3  nearly,  and  this  ratio 
will  always  be  the  same  at  what- 
ever angle  the  ray  enters  the 
medium.  From  air  into  crown 
glass,  the  ratio  is  as  3  to  2  ;  from 
air  into  sulphur,  as  2  to  1 ;  from 
air  into  diamond  as  1  to  f .-  (See 
Fig.  260.) 

By  admitting  the  light  through 
a  small  aperture  at  A,  (Fig.  259,)  so  as  to  pass  through  another 
aperture  at  C,  and  fall  upon  the  bottom  of  the  vessel  at  E,  it  will 
be  found  by  experiment  that  the  three  points  A,  C,  E,  are  al- 
ways in  the  same  plane,  whatever  be  the  angle  of  incidence 
ACP ;  that  is,  the  incident  and  refracted  rays  are  always  in  the 
same  plane. 

747.  Supposing  the  sine  of  the  angle  of  refraction  to  be  al- 
ways 1,  then  the  sine  of  the  angle  of  incidence  will  be  nearly 
1.33  in  water,  and  1.5  in  glass.  The  sine  of  the  angle  of  inci- 
dence, that  of  refraction  being  taken  for  unity,  is  called  the  IN- 
DEX OP  REFRACTION.*  Consequently  it  is  the  ratio  of  the  sine  of 
the  angle  of  incidence  to  that  of  refraction.  Thus  the  index  of 
refraction  for  sulphur  is  2,  because  when  light  passes  out  of  air 
into  sulphur,  the  angle  of  incidence  is  double  that  of  refraction. 
Rays  of  light  which  pass  perpendicularly  out  of  one  medium 
into  another,  suffer  no  refraction  ;  for  the  sine  of  the  angle  of 
incidence  then  becomes  nothing.  When  the  ray  passes  in  the 
opposite  direction,  that  is,  from  a  denser  into  a  rarer  medium,  as 
from  water  into  air,  the  same  constant  ratio  is  found  to  exist  be- 
tween the  sines  of  incidence  and  refraction.  Thus,  (Fig.  259,) 
the  light  from  E  to  C  will  pass  into  CA,  and  the  ratio  of  the 
sines  of  incidence  and  refraction  will  be  that  of  EF  to  AD. 

We  see  an  example  of  the  foregoing  principle  in  the  bent  ap- 


*  It  is  understood  that  the  passage  is  from  air  into'the  given  medium. 


504  NATURAL  PHILOSOPHY. 

pearance  of  an  oar  in  the  water,  the  light  of  the  part  immersed 
(by  which  it  is  visible)  being  turned  from  the  perpendicular,  and 
causing  it  to  appear  higher  than  its  true  place.  In  the  same 
manner,  the  bottom  of  a  river  appears  elevated,  and  diminishes 
the  apparent  depth  of  the  stream.  The  following  ancient  ex- 
periment illustrates  the  same  principle.  If  a  small  piece  of  sil- 
ver be  placed  in  the  bottom  of  a  bowl,  arid  the  eye  be  withdrawn 
until  the  piece  of  silver  disappears,  on  filling  up  the  bowl  with 
water,  the  silver  comes  into  view  again. 

748.  A  ray  of  light  cannot  pass  out  of  a  denser  into  a  rarer  me- 
dium, when  the  angle  of  incidence  is  greater  than  that  at  which  the 
sine  of  the  angle  of  refraction  becomes  equal  to  radius. 

Let  AC  (Fig.  261,)  be  the  ray  incident  upon  tjie  rarer  medium 
RS.  It  will  be  refracted  from  the  perpendicular  DF  into  the  di- 
rection CE,  so  that  AD  is  to  EF  in  a  constant  ratio.  (Art.  746.) 
If  we  increase  the  angle  ACD,  the  angle  FCE  will  also  increase, 
till  the  lines  CE  and  FE  coincide  with  the  Fig.  261. 

radius  CS.  But  if  beyond  this  position  of 
the  ray  AC,  the  angle  ACD  is  still  fur- 
ther increased,  it  is  manifest  its  sine  is 
also  increased  ;  and  consequently,  in  or- 
der that  the  ratio  may  be  constant,  the 
sine  of  refraction  EF  must  also  increase, 
which  is  impossible,  since  it  is  already  by 
hypothesis  equal  to  the  radius  CS.  Hence 
it  follows,  that  whenever  the  angle  of  in- 
cidence is  greater  than  that  at  which  the  sine  of  the  angle  of  re- 
fraction becomes  equal  to  radius,  the  ray  cannot  be  refracted 
consistently  with  the  constant  ratio  of  the  sines. 

This  is  found  to  be  the  case  by  experiment ;  and  at  the  angle 
thus  indicated,  all  the  incident  rays  are  reflected  from  the  inner 
surface  of  the  denser  medium,  having  a  reflexion  more  brilliant 
than  what  can  be  produced  from  any  metallic  surface.  This 
reflexion  is  then  called  total  reflexion. 

In  water,  whose  index  of  refraction  is  1.336,  the  angle  of  total 
reflexion  is  48°  28'.  In  glass,  whose  index  of  refraction  is  1.50, 
it  is  41°  49'.  In  sulphur,  it  is  30°  ;  and  in  diamond,  it  is  23°  35'. 

749.  Transparent  bodies  differ  much  among  themselves  in  re- 
fracting power. 

The  following  table  will  be  useful  by  way  of  reference. 

TABLE   OF   REFRACTIVE   POWERS. 

Index  of  Refraction 

Chromate  of  Lead 2.974 

Red  Silver  Ore, 2.564 

Diamond, 2.439 


OPTICS.  505 

Index  of  Refraction. 

Phosphorus, 2.224 

Sulphur,  (melted.) 2.148 


Glass,  (composed  of  lead  two  parts,  flint  one,) 
Sapphire,  and  other  precious  gems, 

Sulphuret  of  Carbon , 

Oil  of  Cassia, 

Quartz,  or  Rock  Crystal,. 

Amber, 

Crown  Glass , 

Oil  of  Olives, 

Alum, , 

Fluor  Spar, , 

Mineral  Acids, , 

Alcohol, 

Water, , 

Ice, 

Tabasheer, , 


.830 
.800 
.768 
.641 
.548 
.547 
.530 
.470 
.457 
.434 
.410 
.372 
.336 
.309 
.111 


Hence  it  appears,  that  certain  salts  of  silver  and  lead,  the  dia- 
mond, phosphorus,  and  sulphur,  rank  highest  in  refracting  pow- 
er ;  next  come  the  precious  gems,  and  flint  glass,  containing  a 
large  proportion  of  the  oxide  of  lead,  which  has  a  refracting 
power  considerably  higher  than  crown  glass,  containing  less  me- 
tallic oxide  ;  to  which  succeed  the  aromatic  oils.  Among  trans- 
parent solids,  fluor  spar  is  distinguished  for  its  low  refracting 
powers  ;  but  tabasheer,  a  substance  formed  from  the  concreted 
juice  of  the  Indian  bamboo,  is  more  particularly  remarkable  for 
the  same  property.  Figure  260,  will  convey  an  idea  of  the 
comparative  refractive  properties  of  several  of  these  substances. 

In  the  preceding  table,  the  refractive  powers  of  different  bodies 
are  given  without  any  consideration  of  their  densities  or  specific 
gravities  ;  but  it  is  evident,  that  if  a  body  of  small  specific,  grav- 
ity has  the  same  refractive  power  as  another  body  of  greater 
specific  gravity,  the  former  must  have  a  greater  absolute  action 
upon  light  than  the  latter.  Hence,  in  order  to  measure  the  ab- 
solute refractive  powers  of  bodies,  their  specific  gravities  must 
be  taken  into  the  account.  When  estimated  on  this  principle, 
hydrogen  will  be  found  to  have  the  greatest  refractive  power  of 
all  bodies, — it  being,  according  to  Dr.  Brewster,  equal  to  3.0953 ; 
and  it  is  also  the  most  inflammable  of  all  bodies.  It  was  in  con- 
sequence of  the  high  refractive  properties  of  inflammables,  that 
Sir  Isaac  Newton  expressed  the  opinion  that  the  diamond  is  a 
body  of  this  class,  before  its  chemical  constitution  had  been  dis- 
covered.* 

750.  The  Multiplying  Glass  (Fig.  262)  exhibits  as  many  ima- 

_____ _ r 

*  It  is  now  known  to  consist  of  carbon,  or  pure  charcoal. 
64 


506 


NATURAL    PHILOSOPHY. 


B 


ges  of  a  luminous  object,  as  there  are  sur-  Fig.  262. 

faces  exposed  to  it.  The  candle  at  A,  sends 
rays  to  each  of  the  three  surfaces  of  glass. 
Those  which  fall  on  it  perpendicularly, 
pass  directly  through  the  glass  to  the  eye, 
without  change  of  direction,  and  form  one 
image  in  its  true  place  at  A.  But  the  rays 
which  fall  on  the  two  oblique  surfaces,  have 
their  directions  changed  both  in  entering 
and  in  leaving  the  glass,  (as  will  be  seen 
by  following  the  rays  in  the  figure)  so  as  to  meet  the  eye  in  the  di- 
rections of  B  and  C.  Consequently,  images  of  the  candle  are  form- 
ed, also,  at  both  these  points.  A  multiplying  glass  has  usually  a 
great  many  surfaces  inclined  to  one  another,  and  the  number  of 
images  it  forms  is  proportionally  great. 

751.  The  PRISM  is  an  important  instrument  in  Optics,  especial- 
ly as  it  affords  the  means  of  decomposing  light,  and  enters  into 
the  construction  of  several  optical  instruments.  The  triangular 
prism  is  the  only  one  employed  in  experiments,  and  of  this,  noth 
ing  more  is  essential  than  barely  the  inclination  of  two  plane 
transparent  surfaces  to  one  another.  The  optical  prism,  how- 
ever, is  usually  understood  to  be  a  piece  of  solid  glass,  having 
two  sides  constituted  of  equal  parallelograms,  and  a  third  side, 
called  the  base.  The  line  of  intersection  of  the  two  sides  is 
called  the  edge,  and  the  angle  contained  by  the  sides,  the  refract- 
ing angle  of  the  prism.  A  straight  line,  passing  lengthwise  of 
the  prism,  through  its  center  of  gravity,  and  parallel  to  the  edge, 
is  called  the  axis.  A  section  made  by  a  plane  perpendicular  to 
the  axis,  is  an  isosceles  triangle.  Frequently,  the  three  angles  of 
the  prism  are  made  equal  to  one  another,  each  being  60  degrees.* 

Figure  263,  represents  a  sec- 
tion of  a  prism  ABC,  of  which 
AB  is  the  base,  and  ACB  the  re- 
fracting angle.  DE  is  a  beam 
of  the  sun's  light  falling  oblique- 
ly on  the  first  surface  AC,  where 
one  portion  is  reflected  but  an- 
other portion  transmitted.  The 
latter  portion,  instead  of  passing 
directly  forward  and  forming  an 

*  A  very  convenient  prism  for  common  experiments  may  be  constructed  as  fol- 
lows. Select  two  plates  of  window  glass  of  the  best  quality,  or  better,  two  pieces  of 
looking-glass,  from  which  the  silvering  has  been  removed.  The  plates  may  be  five 
or  six  inches  long,  and  one  and  a  half  or  two  inches  broad.  They  are  to  be  united  at 
their  edges  at  an  angle  of  about  60°,  and  furnished  with  a  tin  case,  which  shall  afford 
the  base  and  the  two  ends,  and  a  covering  for  the  edge.  One  of  the  ends  has  an  ori- 
fice with  a  stopper,  for  the  convenience  of  filling  with  a  fluid,  which  may  be  pure 
water,  or  better,  a  saturated  solution  of  the  sugar  of  lead,  filtered  perfectly  clear. 


Fig.  263. 


H 


OPTICS.  507 

image  of  the  sun  at  H,  is  turned  upward  toward  the  perpendic- 
ular pp',  meeting  the  opposite  surface  CB  in  F,  where  it  is  again 
turned  upward,  from  the  perpendicular  p'p,  in  the  direction  FG, 
carrying  the  image  of  the  sun  from  H  to  G.  If  the  incident  and 
emergent  rays  be  produced  so  as  to  meet  in  I,  the  angle  FIH  is 
called  the  angle  of  deviation. 

752.  By  means  of  the  prism,  the  index  of  refraction  for  differ- 
ent bodies  may  be  found  very  conveniently  from  the  following 
theorem. 

The  index  of  refraction  diminished  by  unity,  is  always  equal  to 
the  angle  of  deviation  divided  by  the  refracting  angle  of  the  prism. 
In  demonstrating  this  proposition  it  is  necessary  to  premise, 
that  when  angles  are  small  their  ratio  is  nearly  that  of  their 
sines  ;  and  since  the  sine  of  the  angle  of  incidence  is  to  that  of 
refraction  as  the  index  of  refraction  to  unity,  (Art.  747,)  there- 
fore, n  being  the  index  of  refraction,  (see  Fig.  263,) 

p'El(=DEp)  ip'EF  :  :  n  :  1  .-.  FEI  :  p'EF  :  :  n—  1  :  1  ; 

also,  p'FI(=GFp)  :  p'FE  :  :  n  :  1  /.  EFI  :^'FE  :  :  n—  1  :  1  ; 

.-.  FEI+EFI  :p'EF+/FE  :  :  n-1  :  1, 

/.  FIH:j9'KF::n-l:l 

But  p'KF  and  ACB  are  equal,  being  each  a  supplement  to  four 
right  angles  in  the  quadrilateral  figure  ECFK.     Therefore, 
FIH  :  ACB  :  :  n-1  :  1  .-.  n-1  xACB=FIH. 
u  FIH 

Hence,  "- 


Now  in  prisms  of  glass,  n=|  ;  therefore,  TTJD  ~  %'  or 

£  ACB  ;  that  is,  the  angle  of  deviation  equals  half  the  refracting 
angle  of  the  prism. 

In  order  to  find  the  index  of  refraction  for  any  solid  substance, 
the  substance  itself  maybe  formed  into  a  prism.  The  refracting 
angle  of  the  prism  being  always  known,  and  the  angle  of  devia- 
tion easily  measured,  the  index  of  refraction  is  readily  found,  by 
dividing  the  latter  angle  by  the  former,  and  adding  one  to  the 
quotient.  If  the  substance  is  of  such  a  nature,  that  it  cannot  be 
fashioned  into  a  prism,  as  a  liquid,  for  example,  it  may  then  be 
introduced  into  the  refracting  angle  of  a  prism  formed  by  two 
plates  of  glass  inclined  to  each  other. 

753.  Wh&i  light  is  transmitted  through  a  medium  bounded  by  plane 
and  parallel  surfaces,  the  incident  and  emergent  rays  are  parallel. 

Let  ABba  (Fig.  264,)  be  the  medium  bounded  by  parallel  sur- 
faces AB,  ab  ;  and  let  DE  be  the  incident  ray  refracted  in  the  di- 

Projections  may  be  attached  to  the  two  ends  to  serve  as  handles  or  as  an  axis,  by 
which  the  prism  may  rest  on  supports.  Instead  of  the  tin  case,  we  may  employ  a 
block  of  hard  wood,  first  formed  into  a  triangular  prism,  and  then  dug  out  so  as  to 
admit  the  plates. 


508 


NATURAL   PHILOSOPHY. 


rection  EF    and  emerging  in  the  direction  FG;  the  ray  FG 

will  be  parallel  to  DE.     Through  the  Fig.  264 

points  E,  F,  draw  the  perpendiculars 

PQ,  RS.     Then,  since   PQ  and  RS 

are  parallel,  the  angle  of  refraction 

QEF  at  the  first  surface,  is  equal  to 

EFR,  the  angle  of  incidence  at  the 

second  surface  ;  but  as  the  ratio  of  the 

sine  of  QEF  to  DEP  is  the  same  as 

that  of  EFR  to  SFG,  (Art.  740,)  the 

angles  DEP  and  SFG  must  bo  equal,  and,  consequently,  their 

complements  AED,  &FG;  and  if  we  add  to  these  the  equal 

angles  AEF,  &FE,  the  whole  angles  DEF,  GFE  will  be  equal, 

and  consequently  the  rays  DE,  FG  parallel.* 

It  is  found  by  experiment  that  when  light  is  transmitted  through 
two  contiguous  mediums,  bounded  by  plane  and  parallel  surfaces, 
the  incident  and  emergent  rays  are  parallel  to  one  another,  f 

754.  Through  a  plane  surface,  if  diverging  rays  pass  out  of  a 
rarer  into  a  denser  medium,  they  are  made  to  diverge  less  than 
before  :  if  out  of  a  denser  into  a  rarer  medium,  to  diverge  more. 

For  since  the  sine  of  the  angle  of  refraction  is  always  as  that 
of  incidence,  the  most  divergent  lines  in  a  pencil  will  be  the  most 
refracted,  and  will  of  course  be  brought  nearer  to  a  parallelism 
with  those  rays  which  diverge  less  when  the  refraction  is  toward 
the  perpendicular,  but  will  be  still  further  separated  when  the 
refraction  is  from  the  perpendicular. 

755.  LENSES,  on  account  of  their  extensive  use  in  the  construc- 
tion of  optical  instruments,  require  very  particular  attention  in 
the  study  of  Optics.     They  are  of  several  varieties,  as  is  shown 
in  the  following  figure. 

Fig.  265. 


ED  C  B  A 

A  double  convex  lens  (A)  is  a  solid  formed  by  two  segments  oi 
a  sphere,  base  to  base.J 

A  plano-convex  lens  (B)  is  a  lens  having  one  of  its  sides  convex 
and  the  other  plane,  being  simply  a  segment  of  a  sphere. 

*  Euc.  I,  27.  t  Wood's  Optics,  p.  40. 

t  Though  this  is  the  most  common  form  of  the  double  convex  lens,  yet  it  is  not 
essential  that  the  two  segments  should  be  portions  of  the  same  sphere  :  they  may  be 
segments  of  different  spheres,  in  which  case  the  curvatures  will  be  unequal  on  the 
two  sides  of  the  lens. 


OPTICS.  509 

A  double  concave  lens  (C)  is  a  solid  bounded  by  two  concave 
spherical  surfaces,  which  may  be  either  equally  or  unequally 
concave. 

A  plano-concave  lens  (D)  is  a  lens  one  of  whose  surfaces  is 
plane  and  the  other  concave. 

A  meniscus  (E)  is  a  lens  one  of  whose  surfaces  is  convex  and 
the  other  concave,  but  the  concavity  being  less  than  the  convex- 
ity, it  takes  the  form  of  a  crescent,  and  has  the  effect  of  a  convex 
lens  whose  convexity  is  equal  to  the  difference  between  the  sphe- 
ricities of  the  two  sides. 

A  concavo-convex  lens  (F)  is  a  lens  one  of  whose  surfaces  is 
convex  and  the  other  concave,  the  concavity  exceeding  the  con- 
vexity, and  the  lens  being  therefore  equivalent  to  a  concave  lens, 
whose  sphericity  is  equal  to  the  difference  between  the  spherici- 
ties of  the  two  sides. 

A  line  (MN)  passing  through  the  center  of  a  lens  perpendicular 
to  its  opposite  surfaces,  is  called  the  axis. 

756.  The  manner  in  which  light  is  refracted  by  passing  into 
denser  or  rarer  mediums  bounded  by  spherical  surfaces,  may  be 
readily  understood  and  easily  remembered,  by  keeping  in  mind 
the  position  of  the  incident  rays  with  respect  to  the  perpendicu- 
lar, that  is,  the  radius  of  the  spherical  surface.  Suppose  the  two 
mediums  are  air  and  glass,  and  let  us  take  first,  the  case  of  a  con- 
vex surface  of  glass  :  then,  since  rays  passing  into  the  glass  would 
be  turned  toward  the  perpendiculars,  (all  of  which  being  radii, 
tend  toward  a  common  center,)  parallel  rays  would  be  made  to 
converge;  diverging  rays  would  become  less  diverging;  converg- 
ing rays,  more  converging.  These  are  the  general  results ;  but 
let  us  trace  the  progress  of  diverging  and  converging  rays  a  little 
more  particularly.  If  the  rays  came  from  a  near  radiant,  so  as 
to  diverge  very  much  from  each  other,  the  effect  of  the  glass 
would  be  simply  to  diminish  their  divergency  ;  but  if  they  came 
from  some  more  distant  point,  so  as  to  be  less  diverging,  they 
might  be  turned  so  far  toward  the  perpendicular  as  to  become 
parallel,  or  even  converging.  But  suppose  the  incident  rays  to 
come  to  the  glass  converging,  then  if  they  were  directed  toward 
the  center  of  the  sphere  they  would  coincide  with  the  radii  or 
perpendiculars  and  suffer  no  change  of  direction  ;  if  they  origin- 
ally tended  to  a  point  more  distant  than  the  center,  being  turned 
toward  the  radii,  they  would  be  rendered  more  convergent ;  but 
if  they  tended  toward  a  point  nearer  than  the  center,  for  the 
same  reason  they  will  converge  less  than  before. 

These  several  cases  will  be  rendered  familiar  by  studying  the 
representation  in  Fig.  266.* 

*  The  student  is  expected  to  make  the  explanation  of  each  case  from  the  figure, 
following  the  rays  AN,  &c.  to  GN.  Thus,  AN  being  refracted  toward  the  perpen- 


510 


NATURAL   PHILOSOPHY. 
Fig.  266. 


M 


G'      C'      F 


757.  Secondly,  let  us  consider  the  case  of  a  concave  surface 

We  shall  perceive,  by  inspecting  Fig.  267,  that  parallel  rays,  b 

Fig.  267. 


A      M 

being  turned  toward  the  perpendicular,  are  made  diverging ;  di- 
verging rays  are,  in  general,  rendered  more  diverging  ;  but  when 
they  come  from  the  center  of  concavity,  they  suffer  no  refraction, 
and  when  from  a  point  nearer  the  surface  than  the  center,  they 
diverge  less  than  before  ;  and  converging  rays  are,  in  general, 
rendered  less  converging,  but  they  may  be  so  slightly  convergent 
before,  that  the  refracting  power  of  the  glass  shall  be  sufficient 
to  render  them  parallel  or  even  divergent. 

758.  Thirdly,  if  we  now  trace  the  progress  of  the  rays  through 
LENSES,  we  shall  readily  follow  their  course  by  applying  the  fore- 
going principles. 

1.  Let  AB  (Fig.  268,)  be  a  double  convex  lens,  C,  C',  the  cen- 
ters of  curvature,  and  ED  a  ray  of  light  falling  upon  the  lens 
at  D.  According  to  the  principles  just  explained,  ED  would  be 
.turned  toward  CD,  the  perpendicular  to  the  refracting  surface, 


dicular  NC',  is  rendered  less  diverging ;  BN  is  turned  so  far  toward  NC'  as  to  becom« 
parallel  to  AM,  &c. 


OPTICS. 


511 


and  instead  of  passing  onward  in  the  same  straight  line  EDG,  it 
would  proceed  in  the  line  DD'.  "  Again,  on  passing  out  of  the 
denser  into  the  rarer  medium  at  the  second  surface  at  D',  instead 
of  proceeding  onward  in  the  line  DD'H,  it  would  be  turned  fur- 
ther from  the  perpendicular  to  that  surface,  namely  C'D',  so  as 
to  proceed  in  the  line  D'F.  Both  surfaces  of  the  lens,  therefore, 
conspire  to  turn  the  ray  out  of  its  former  course,  and  when  the 
curvature  of  the  two  sides  is  the  same,  they  contribute  equally 
to  produce  this  effect. 


2.  Let  AB  (Fig.  269,)  be  a  double  concave  lens,  then  by  tra- 
cing the  progress  of  the  i*ay  ED,  DD',  D'F,  it  will  be  seen  that 
the  effect  of  each  surface  of  the  lens  is  to  cause  the  ray  to  di- 
verge further  from  the  axis.  Thus  C'D,  CD',  being  the  radii  of 
curvature,  the  ray  ED,  on  entering  the  lens,  is  refracted  into  the 
line  DH  ;  and  again,  on  leaving  the  lens,  it  is  refracted  into  D'F. 

759.  In  a  double  convex,  or  double  concave  lens,  there  is  a  cer- 
tain point  called  its  center,  through  which  every  ray  that  passes, 
has  its  incident  and  emergent  rays  parallel. 


Fig.  270. 


Let  R,  r,  (Figs.  270,  271,)  be  the  centers  from  which  the  sur- 
faces of  these  lenses  are  described,  and  REr  their  axis.  Draw 
any  two  of  their  radii  RA,  ra,  parallel  to  each  other,  and  join 
Aa  ;  the  point  E,  where  this  line  intersects  the  axis,  will  be  the 
'point  above  described,  and  any  ray,  as  Qq,  passing  through  A, 
will  have  the  incident  ray  QA,  parallel  to  the  emergent  ray  aq. 


512  NATURAL  PHILOSOPHY. 

Fig.  271. 


For  since  the  triangles  REA,  rEa,  are  similar,  RA  :  ra  : :  RE  : 
rE,  .'.  RAitra  :  ra  : :  RErbrE  :  rE.  And,  as  the  three  first  terms 
of  this  proportion  are  invariable,  the  last,  rE,  must  also  be  inva- 
riable. Hence  it  follows,  that  to  whatever  points  in  the  surface 
of  the  lens,  the  parallel  radii  RA,  ra,  are  drawn,  the  line  Aa  will 
always  cut  the  axis  Rr  in  the  same  point  E.  If  we  now  sup- 
pose the  ray  Aa  to  pass  both  ways  out  of  the  lens,  it  will  be  re- 
fracted equally  and  in  contrary  directions  ;  because  RA,  ra,  being 
perpendiculars  to  the  surface  at  A  and  a,  the  angles  of  incidence 
of  the  ray  Aa  or  aA,  will  be  equal.  Consequently,  AQ.  will  be 
parallel  to  aq.  When  the  thickness  of  the  lens  is  inconsiderable, 
and  when  a  ray  falls  nearly  perpendicularly  upon  it,  the  part  of 
the  ray  through  E,  viz.  QAEa</,  may  be  taken  as  a  straight  line, 
passing  through  the  center  E  of  the  lens  ;  for  the  perpendicular 
distance  between  AQ,  aq,  diminishes,  both  with  the  thickness  of 
the  lens  and  with  the-  obliquity  of  the  ray  to  the  axis. 

760.  The  office  of  a  convqx  lens  is  to  collect  rays  of  light. 
Hence,  when  applied  to  parallel  rays,  it  makes  them  converge  ; 
to  diverging  rays,  it  makes  them  diverge  less ;  and  to  converging 
rays,  it  makes  them  converge  more.  Moreover,  with  regard  to 
diverging  rays,  the  degree  of  divergence  may  be  reduced  so 
much  as  to  render  the  rays  parallel,  or  even  to  make  them  con- 
verge, which  will  depend  both  on  the  position  of  the  radiant,  as 
illustrated  in  Art.  756,  and  on  the  power  of  the  lens. 

On  the  contrary,  the  office  of  a  concave  lens  is  to  separate  rays 
of  light.  Hence,  when  it  is  applied  to  parallel  rays,  it  makes 
them  diverge  ;  to  rays  already  diverging,  it  makes  them  diverge 
more  ;  and  to  converging  rays,  it  makes  them  converge  less,  be- 
come parallel,  or  even  diverging.* 

With  these  general  principles  in  view,  we  may  now  advan- 
tageously investigate  the  manner  in  which  IMAGES  are  formed 
by  means  of  lenses. 

1.  If  we  place  a  radiant,  as  a  candle,  nearer  to  a  lens  than  its 


•  A  striking  analogy  will  be  remarked  between  the  convex  lens  and  concave  mirror, 
and  between  the  concave  lens  and  convex  mirror. 


OPTICS.  513 

principal  focus,  then,  since  the  rays  go  out  diverging,  (Art.  756,) 
no  image  will  be  formed  on  the  other  side  of  the  lens. 

2.  If  we  place  the  radiant  in  the  focus,  the  rays  will  go  out 
parallel,  but  will  still  not  be  collected  into  a  distinct  image. 

3.  If  the  radiant  is  removed  further  from  the  lens  than  its 
principal  focus,  then  the  rays  will  be  collected  on  the  other  side 
of  the  lens,  so  as  to  form  a  distinct  representation  of  the  object. 

As  this  last  case  is  particularly  important,  since  it  exhibits  the 
manner  in  which  images  are  formed  by  means  of  convex  lenses, 
let  us  examine  it  with  more  attention. 


761.  Rays  of  light  diverging  from  the  several  points  of  any  ob- 
ject, which  is  further  from  a  convex  lens  than  its  principal  focus, 
will  be  made  to  converge  on  the  other  side  of  the  lens,  to  points  cor- 
responding to  those  from  which  they  diverged,  and  will  form  an 
image. 

Let  MN  (Fig.  272,)  be  a  luminous  object  placed  before  a  dou- 
ble convex  lens  LL.  Now  every  point  in  the  radiant  sends  forth 
innumerable  rays  in  all  directions,  part  of  which  fall  upon  the 
lens  LL.  Each  pencil  may  be  considered  as  a  cone  of  rays, 
having  for  its  axis  the  straight  line  which  passes  through  the 
center  of  the  lens,  which  line  suffers  no  change  of  direction,  (Art. 
757,)  while  those  rays  of  the  pencil  which  strike  upon  the  ex- 
treme parts  of  the  lens,  form  the  exterior  rays  of  the  cone  ;  all 
the  others  are  of  course  included  between  these.  It  will  be  suf- 
ficient to  follow  the  course  of  the  central  and  the  two  extreme 
rays.  Let  ML,  MC,  ML,  represent  such  a  pencil.  The  two  ex- 
treme rays  will  be  collected  by  the  lens  and  made  to  meet  in  the 
axis  or  central  ray,  in  some  point  on  the  other  side,  as  at  m.  For 
the  same  reason,  every  other  point  in  the  object  will  have  its  cor- 
responding point  in  the  image,  and  all  these  points  of  the  image 
taken  together,  will  form  a  true  representation  of  the  object.  By 
inspecting  the  figure,  it  will 

be  seen,  that  the  axes  of  all  Jf 

the  pencils  cross  each  other 
in  the  center  of  the  lens ; 
that  the  part  corresponding 
to  the  top  of  the  object  is 
carried  to  the  bottom  of  the 
image,  while  that  corres- 
ponding to  the  bottom  of  the 
object  is  at  the  top  of  the 
image,  and,  consequently,  that  the  image  is  inverted  with  re^ 
spect  to  the  object.  It  will  be  further  seen,  that  although  the  in- 
dividual rays  which  make  up  a  single  pencil  are  made,  on  pass- 
ing through  the  lens,  to  converge,  yet  the  axes  of  all  the  pencils 
go  out  diverging  from  each  other,  which  carries  them  further 

65 


514  NATURAL   PHILOSOPHY. 

and  further  asunder,  the  further  they  proceed  before  they  come 
to  a  focus. 

762.  The  diameter  of  the  object  is  to  the  diameter  of  the  image  as 
the  distance  of  the  object  from  the  lens  is  to  the  distance  of  the  image 
from  the  lens. 

For  the  two  triangles  MOC  and  moC  are  similar;  therefore 
MO :  mo  : :  CO  :  Co.  With  a  given  object,  the  diameter  of  the 
image  is  as  its  distance  from  the  lens.  And,  since  the  surfaces 
of  the  object  and  the  image  are  similar  figures,  (being  parallel 
sections  of  similar  pyramids,  or  cones,  whose  vertices  meet  in  the 
center  of  the  lens,)  the  surface  of  the  image  is  as  the  square  of 
its  distance  from  the  lens.  By  bringing  the  object  nearer  to  the 
lens,  the  image  recedes  from  it  on  the  other  side,  since  the  rays, 
being  more  divergent,  are  not  so  soon  brought  to  a  focus ;  there- 
fore, by  bringing  the  radiant  very  near  to  the  focus  of  parallel 
rays,  so  as  to  throw  the  image  very  far  back,  the  latter  becomes 
exceedingly  magnified. 

The  diameter  of  the  image  will  not  be  altered  by  changing  the 
area  of  the  lens ;  for  that  diameter  will  be  determined  in  all 
cases  by  the  distances  between  the  axes  of  the  two  pencils  which 
come  from  the  extremities  of  the  object,  and  cross  each  other  in 
the  center  of  the  lens.  The  size  of  the  image,  however,  will  be 
affected  by  changing  the  convexity  of  the  lens,  while  the  object 
remains  the  same  and  at  the  same  place,  being  found  nearer  the 
lens,  as  the  latter  is  more  convex. 

763.  Rays  proceeding  from  any  radiant  point,  which  are  refract- 
ed by  the  different  parts  of  the  same  lens,  do  not  meet  accurately  in 
one  focus,  but  their  points  of  meeting  are  spread  over  a  certain  space, 
whose  diameter  is  called  the  SPHERICAL  ABERRATION  of  the  lens. 

Let  LL  (Fig.  273)  be  a  plano-convex  lens,  on  which  are  in 
cident  the  parallel  rays  RL,  RL,  at  the  extremities,  and  R'L', 
R'L',  near  the  axis :  then,  according  to  Art.  756,  the  axis  will 
proceed  on  without  any  change  of  direction,  and  the  rays  which 
are  very  near  to  the  axis,  being  also  nearly  perpendicular  to  the 
refracting  surface,  sustain  only  a  slight  change  of  direction,  suf- 
ficient, however,  to  collect  Fig.  273. 
them  into  a  focus  at  some 
distance  from  the  lens  in  the 
point  F.  But  the  rays  RL, 
RL,  meeting  the  refracting 
surface  more  obliquely,  are 
more  turned  out  of  their 
course,  and  are  therefore  R 
collected  into  a  focus  in  some  point  nearer  to  the  lens  than  F,  as 
at/.  The  intermediate  rays  refracted  by  the  lens  will  have 


OPTICS.  515 

their  foci  between  F  and  /.  Continue  the  lines  L/  and  L/,  till 
they  meet  at  G  and  H,  a  plane  passing  through  F.  The  distance 
fF  is  called  the  longitudinal  spherical  aberration,  and  GH  the 
lateral  spherical  aberration. 

It  is  obvious  that  such  a  lens  cannot  form  a  distinct  picture  of 
any  object  in  its  focus  F.  If  it  is  exposed  to  the  sun,  the  central 
parts  of  the  lens  L'mL',  whose  focus  is  at  F,  will  form  a  pretty 
bright  image  of  the  sun  at  F ;  but  as  the  rays  of  the  sun  which 
pass  through  the  outer  part  LL  of  the  lens  have  their  foci  at 
points  between  /  and  F,  the  rays  will,  after  arriving  at  these 
points,  pass  on  to  the  plane  GH,  and  occupy  a  circle  whose  diam- 
eter is  GH ;  hence  the  image  of  the  sun  in  the  focus  F,  will  be 
a  bright  disk,  surrounded  and  rendered  indistinct  by  a  broad  halo 
of  light,  growing  fainter  and  fainter  from  F  to  G  and  H.  In  like 
manner,  every  object  seen  through  such  a  lens,  and  every  image 
formed  by  it,  will  be  rendered  confused  and  indistinct  by  sphe- 
rical aberration. 

If  we  cover  up  all  the  exterior  portions  of  the  lens,  so  as  to 
permit  only  those  portions  of  the  rays  which  lie  near  the  axis  to 
pass  through  the  lens,  then  the  rays  all  meet  at  or  very  near  to 
the  point  F,  and  a  much  more  distinct  image  is  formed ;  but  so 
much  of  the  light  is  excluded  by  this  process,  that  the  brightness 
of  the  image  is  considerably  diminished.  The  dimensions  of  the 
image  are  the  same  in  both  cases.  (Art.  762.) 

764.  By  experiments  made  with  different  kinds  of  lenses,  the 
following  results  are  obtained.  In  plano-convex  lenses,  placed  as 
in  Fig.  272,  the  greatest  spherical  aberration  is  4?  times  mn,  the 
thickness  of  the  lens.  In  a  plano-convex  lens,  with  its  convex 
side  turned  toward  the  parallel  rays,  the  aberration  is  only 
1  TWths  of  its  thickness.  In  using  the  plano-convex  lens,  there- 
fore, it  should  always  be  so  placed,  that  the  parallel  rays  may  be 
incident  upon  the  convex  surface.  In  a  double  convex  lens,  with 
equal  convexities,  the  aberration  is  1  r\\  ths  of  its  thickness.  The 
lens  which  has  the  least  spherical  aberration,  is  a  double  convex 
one,  whose  radii  are  as  1  to  6.  When  the  face,  whose  radius  is 
1,  is  turned  toward  the  parallel  rays,  the  aberration  is  only 
1  yj^ths  °f  its  thickness.  Hence,  the  lenses  employed  in  optical 
instruments  are  made  very  thin  ;  and  the  light  is  suffered  to  pass 
only  through  the  central  parts  of  the  lens.  As  the  central  parts  of 
the  lens  LL,  refract  the  rays  too  little,  and  the  marginal  parts 
too  much,  it  is  evident,  that  if  we  could  increase  the  convexity 
at  n,  and  diminish  it  gradually  toward  L,  we  should  remove  the 
spherical  aberration.  But  the  ellipse  and  hyperbola  are  curves 
of  this  kind,  in  which  the  curvature  diminishes  from  n  to  L ;  and 
mathematicians  have  shown  how  spherical  aberration  may  be 
entirely  removed,  by  lenses  whose  sections  are  ellipses  or  hyper- 
bolas. Of  a  lens  of  this  kind  we  will  annex  one  example. 


516  NATURAL   PHILOSOPHY. 

765.  A  lens  in  the  form  of  a  spheroid,  (generated  by  the  revolu 
tion  of  an  ellipse  about  its  major  axis,)  whose  major  axis  is  to  the 
distance  between  its  foci,  as  the  sine  of  incidence  to  the  sine  of  re- 

fraction,  will  cause  parallel  rays  incident  in  the  direction  of  its 
axis,  to  converge  accurately  to  the  remoter  focus. 

Let  BDK,  (Fig.  274,)  be  the  generating  ellipse,  H  and  I  its 
foci ;  then,  by  the  supposition, 

DK  :  HI :  :  sin.  Incidence  :  sin.  Refraction. 

Let  AB,  which  is  paral-  fig.  374. 

lei  to  DK,  be  a  ray  of  light 
incident  upon  the  spheroid. 
Join  HB,  IB  ;  draw  EEC, 
touching  the  generating 
ellipse  in  B ;  through  B 
and  H.  draw  GBL  and  HCO 
at  right  angles  to  EBC  ;  let 
GBL  meet  DK  in  N;  and 
produce  IB  till  it  meets 
HCO  in  O.  Then,  since 
HBC=1BE,*  and  OBC=IBE,  therefore  HBC=OBC.  Also,  BCH, 
BCO,  are  right  angles,  and  BC  is  common  to  the  two  triangles 
BCH,  BCO  ;  therefore  BO=BH,  and  IO=DK  ;  consequently, 

IO  :  IH  :  :  sin.  Incidence  :  sin.  Refraction.  And  because  BN  is 
parallel  to  OH, 

IB  :  IN  :  :  IO  :  IH  :  :  sin.  Incid.  :  sin.  Refrac. 

Also,  IB  :  IN  : :  sin.  INB :  sin.  IBN  : :  sin.  BNH  or  sin.  ABG  :  sin. 
IBL ;  therefore,  sin.  ABG  :  sin.  IBL  :  :  sin.  Incid.  :  sin.  Refrac. 
And  since  sin.  ABG  is  the  sine  of  incidence,  sin.  IBL  is  the  sine 
of  refraction,  LBI  is  the  angle  of  refraction,  and  BI  is  the  refract- 
ed ray.  In  the  same  manner  it  may  be  shown,  that  every  other 
ray  in  the  pencil  will  be  refracted  to  I. 

766.  If  from  the  center  I,  Fig.  275. 
(Fig.  275,)  with  any  radius 

less  than  ID,  a  circular  arc 
PQ  be  described,  the  solid, 
generated  by  the  revolution 
of  PDQ  about  the  axis  DI, 
will  refract  all  the  rays  inci- 
dent parallel  to  DI,  accurate- 
ly to  I.  For,  after  refraction 
at  the  surface  PDQ,  the  rays 
converge  to  I ;  and  they  suf- 
fer no  refraction  at  the  surface  PQ,  because  they  are  incident 
perpendicularly  upon  it.f 

Hence  it  follows  that  a  meniscus,  whose  convex  surface  is  part 

•  Conic  Sections.  t  Wood's  Optics,  Sec.  187, 188. 


OPTICS.  517 

of  an  ellipsoid,  and  whose  concave  surface  is  part  of  any  spherical 
surface  whose  center  is  in  the  farther  focus,  may  be  so  construct- 
ed as  to  have  no  spherical  aberration,  and  to  refract  parallel 
rays  incident  on  its  convex  surface,  to  the  farther  focus.  When 
the  foregoing  properties  of  the  ellipse  were  discovered,  (and  sim- 
ilar properties  belong  to  the  hyperbola,)  philosophers  exerted  all 
their  ingenuity  in  grinding  and  polishing  lenses  with  elliptical 
and  hyperbolical  surfaces ;  and  various  ingenious  mechanical 
contrivances  were  proposed  for  this  purpose.  These,  however, 
have  not  succeeded  ;  and  the  difficulty  of  grinding  glasses  of  any 
other  than  a  spherical  curvature,  is  such  as  to  prevent  the  use 
of  spheroidal  and  other  forms  not  subject  to  aberration ;  but 
other  expedients  have  been  devised  for  correcting  this  error. 

Thpugh  we  cannot  remove  or  diminish  the  spherical  aberra- 
tion of  single  lenses  beyond  lT|^ths  of  their  thickness,  yet  by 
combining  two  or  more  lenses,  and  making  opposite  aberrations 
correct  each  other,  we  can  remedy  this  defect  to  a  very  consider- 
able extent  in  some  cases,  and  in  other  cases  remove  it  alto- 
gether.* The  manner  in  which  this  is  effected,  will  be  more 
particularly  pointed  out  in  connection  with  the  subjects  of  Mi- 
croscopes and  Telescopes. 


CHAPTER  V. 

OF  THE  DECOMPOSITION  OF  LIGHT  AND  THE  SOLAR 
SPECTRUM.t— NATURE  OF  LIGHT. 

767.  IN  tracing  the  course  of  rays  of  light  through  a  refract- 
ing medium,  we  have  thus  far  supposed  them  to  be  homogene- 
ous, and  to  be  all  affected  in  the  same  manner.  But  in  nature, 
the  fact  is  otherwise  ;  that  is, 

The  sun's  light  consists  of  rays  which  differ  in  refrangibility  and 
in  color. 

The  glass  prism,  in  consequence  of  the  strong  refraction  of 
light  which  it  produces,  (see  Art.  751,)  is  well  fitted  for  experi- 
ments of  this  kind.  We  procure,  therefore,  a  triangular  prism 
of  good  flint  glass,  and  having  darkened  a  room,  admit  a  sun- 
beam obliquely,  through  a  small  round  hole  in  the  window  shut- 
ter. Across  this  beam,  near  the  shutter,  we  place  the  prism, 
with  its  edge  parallel  to  the  horizon,  so  as  to  receive  the  beam 
upon  one  of  its  sides.  The  rays,  on  passing  through  the  prism, 
will  be  refracted  and  thrown  upward,  as  will  be  rendered  evi- 
dent by  conceiving  perpendiculars  drawn  to  the  surface  of  the 

*  Brewster. 

t  That  part  of  Optics  which  treats  of  colors,  is  sometimes  denominated  Chromatic*. 


518 


NATURAL   PHILOSOPHY. 


prism  at  the  points  of  incidence  and  emergence.  If  now  we  re- 
ceive the  refracted  rays  upon  a  screen,  at  some  distance,  they 
will  form  an  elongated  image,  exhibiting  the  colors  of  the  rain- 
bow, namely,  red,  orange,  yellow,  green,  blue,  indigo,  violet,  to- 
gether composing  the  prismatic  spectrum.  (See  Fig.  276.) 
Fig.  276. 


S,  a  sunbeam. 

F,  a  hole  in  the  window  shutter. 

ABC,  the  prism,  having  its  refracting  angle  ACB  downward. 
Y,  a  white  spot,  being  an  image  of  the  sun  formed  on  the 
floor  before  the  prism  is  introduced. 

MN,  the  screen  containing  the  spectrum.* 

768.  On  viewing  the  spectrum  attentively,  we  perceive  that 
the  lowest  or  least  refracted  extremity  is  a  brilliant  red,  more 
full  and  vivid  than  can  be  produced  by  any  other  means,  or  than 
the  color  of  any  natural  substance.  This  dies  away,  first  into 
an  orange,  and  then  passes  by  imperceptible  gradations  into  a 
fine  pale  straw-yellow,  which  is  quickly  succeeded  by  a  pure  and 
very  intense  green,  which  again  passes  into  a  blue,  at  first  of 
less  purity,  being  mixed  with  green,  but  afterward,  as  we  trace 
it  upward,  deepening  into  the  purest  indigo.  Meanwhile,  the  in- 
tensity of  the  illumination  is  diminishing,  and  in  the  upper  por- 
tions of  the  indigo  tint,  it  is  very  feeble  ;  but  the  blue  is  continued 
still  beyond,  and  acquires  a  pallid  cast  of  purplish  red,  a  livid 
hue  better  seen  than  described,  and  which,  though  not  to  be  ex  • 
actly  matched  by  any  natural  color,  approaches  most  nearly  to 
that  of  a  fading  violet,  f 

A  pleasing  way  of  exhibiting  the  separate  colors  of  the  spec- 
trum, is  to  throw  the  prismatic  beam  on  a  distant  wall  or  screen, 

*  The  opposite  white  wall  of  plaster  or  stucco,  may  serve  the  purpose  of  a  screen  ; 
or  the  screen  may  be  made  of  a  large  sheet  of  white  paper ;  but  a  convenient  screen 
for  the  lecture  room  is  made  by  pasting  a  large  sheet  of  muslin  to  a  frame,  and  at. 
taching  it  to  a  movable  stand.  If  the  cloth  is  thick,  it  may  be  wet 

t  Herschel  on  Light. 


OPTICS.  519 

so  as  to  form  a  long  spectrum,  and  into  this  beam,  at  some  con- 
venient distance  from  the  prism,  as  near  A,  (Fig.  276,)  to  intro- 
duce a  concave  lens  of  a  size  sufficient  to  cover  each  of  the  dif- 
ferent colored  pencils  successively.  The  lens  will  cause  the  rays 
of  the  same  color  to  diverge,  and  to  form  a  circular  image  on 
the  screen,  which  will  distinguish  them  very  strikingly  from  the 
contiguous  portions  of  the  spectrum. 

769.  If  rays  of  the  same  color  in  the  prismatic  beam  be  insulated 
from  the  rest,  and  made  to  pass  through  a  second  prism,  they  are 
refracted  as  usual,  (the  amount  of  refraction  being  different  for  the 
different  colored  rays,)  but  they  undergo  no  further  change  of  color. 

To  perform  this  experiment,  we  provide  a  board,  perforated 
with  a  small  round  hole,  and  mounted  on  a  stand.  This  screen 
is  placed  across  the  prismatic  beam,  a  little  way  from  the  prism, 
in  such  a  manner  as  to  permit  rays  of  the  same  color  only  to 
pass  through  the  aperture,  while  the  other  portions  of  the  beam 
are  intercepted.  The  homogeneous  light  thus  insulated  is  made 
to  pass  through  a  second  prism,  and  its  image  is  thrown  on  the 
Fig.  277. 


e  E 

wall.  The  experiment  will  be  more  perfect,  if  the  homogeneous 
pencil  is  made  to  pass  through  a  second  screen  similar  to  the 
first,  so  as  to  let  only  the  central  rays  fall  upon  the  second  prism. 
This  second  refraction  produces  no  change  of  color.  It  will  be 
found,  however,  that  while  all  other  things  remain  the  same,  the 
several  images  formed  of  homogeneous  rays,  will  occupy  differ- 
ent positions  on  the  wall,  the  red  being  lowest,  the  violet  high- 
est, and  the  intermediate  colors  will  be  arranged  between  them 
in  the  order  of  their  refrangibilities.  (See  Fig.  277.) 

In  addition  to  the  parts  of  the  figur^  enumerated  in  Fig.  277, 
DE  represents  the  first  screen,  which  permits  only  one  sort  of 
rays  to  pass  by  a  small  aperture  at  G,  and  de  represents  a  second 
screen,  which  permits  only  the  central  rays  of  this  pencil  to  pass 
by  a  small  hole  at  g  ;  abc  is  the  second  prism,  and  M  is  the  im- 
age of  homogeneous  light  on  the  wall. 

770.  The  light  of  the  sun  reflected  from  the  first  surface  of  bodies, 
and  also  the  white  Jlames  of  all  combustibles,  whether  direct  or  re- 
jlected,  differ  in  color  and  refrangibility,  like  the  direct  light  of  the 
sun. 


520  NATURAL   PHILOSOPHY. 

The  truth  stated  in  this  proposition  was  established  by  New- 
ton, by  experiments  with  the  prism,  similar  to  those  detailed  in 
connection  with  the  preceding  proposition. 

771.  The  sun's  light  is  compounded  of  all  the  prismatic  colors, 
mixed  in  due  proportion. 

If  we  collect,  by  means  of  a  convex  lens,  the  different  colored 
pencils  in  the  prismatic  beam,  just  after  they  have  emerged  from 
the  prism,  (see  Fig.  276,)  the  image  formed  by  the  lens  will  be 
perfectly  white.  A  concave  mirror  may  be  used  instead  of  the 
lens,  the  image  being  thrown  on  a  screen.  Or  the  rays  after 
they  have  passed  the  prism  may  be  received  on  a  second  prism 
of  the  same  kind,  placed  near  the  first,  but  with  its  refracting 
angle  in  the  opposite  direction.  In  this  case  the  second  prism 
restores  the  light  to  its  usual  whiteness. 

That  all  the  different  colors  of  the  spectrum  are  essential  to 
the  composition  of  white  light,  may  be  rendered  evident  by 
intercepting  a  portion  of  any  one  of  the  colors  of  the  spectrum 
before  they  have  been  reunited  as  in  the  foregoing  experiments. 
Thus,  if  we  introduce  a  thread  or  a  wire  into  any  part  of -the  pris- 
matic beam  between  the  prism  and  the  lens,  the  image  formed 
by  the  lens  will  be  no  longer  white  but  discolored.  If,  instead  of 
the  wire,  an  instrument,  shaped  like  a  comb  with  coarse  broad 
teeth,  be  introduced  into  the  beam,  the  discoloration  of  the  im- 
age is  more  diversified,  the  colors  of  the  image  being  those  com- 
pounded of  the  prismatic  colors,  which  are  not  intercepted  by 
the  comb.  If  the  teeth  of  the  comb  be  passed  slowly  over  the 
beam,  a  succession  of  different  colors  appears,  such  as  red,  yel- 
low, green,  blue,  and  purple  ;  but  if  the  motion  of  the  comb  be 
rapid,  all  these  different  hues  become  blended  into  one  by  the 
momentary  continuance  of  each  in  the  eye,  and  the  sensation  is 
that  of  white  light. 

For  a  similar  reason,  if  the  colors  of  the  spectrum  are  painted 
on  a  top,  in  due  intensity  and  proportion,  and  the  top  be  set  to 
spinning,  the  sensation  will  be  that  of  white  light.  Or  the  colors 
of  the  spectrum  may  be  first  laid  on  a  sheet  of  paper,  and  this 
may  be  pasted  on  a  cylinder  of  wood,  which  may  be  made  to  re- 
volve on  the  whirling  tables  :  the  result  will  be  the  same.  New- 
ton tried  various  experiments  with  different  colored  powders, 
grinding  together  such  as  corresponded  as  nearly  as  possible  to 
the  colors  of  the  spectrum.  By  these  means  he  was  able  to  pro- 
duce, from  the  mixture  of  seven  different  colored  powders,  a 
grayish-white,  but  could  never  reach  a  perfectly  clear  white,  ow- 
ing to  the  difficulty  of  finding  powders  whose  colors  correspond 
exactly  to  those  of  the  spectrum. 

772.  Several  of  the  colors  of  the  spectrum  may  be  produced  by 
the  mixture  of  other  colors  ;  as  green  by  the  mixture  of  yellow  and 
blue,  orange  by  red  and  yellow,  fyc. 


OPTICS.  521 

Experiments  were  devised  by  Newton  for  thus  combining  the 
colors  of  two  contiguous  spectrums,  transferring,  for  example, 
the  blue  of  one  to  the  yellow  of  the  other,  and  forming  green  by 
their  union.  On  causing  this  compound  green,  however,  to  pass 
through  the  prism,  it  is  resolved  into  its  original  colors,  yellow 
and  blue,  whereas,  the  green  of  the  spectrum  is  not  thus  resolved 
by  the  prism.  Hence,  Newton  infers  that  the  green  of  the  spec- 
trum is  not  a  compound  but  a  simple  original  color,  and  so  of  all 
the  rest. 

It  has,  however,  been  a  question  among  opticians,  since  the 
time  of  Newton,  what  is  the  number  of  original  or  fundamental 
colors  in  the  spectrum  ?  Many  years  since,  Mayer  advanced  the 
hypothesis  that  the  only  simple  colors  in  the  solar  spectrum  are 
red.  yellow,  and  blue, — all  the  others  being  compounded  of  these  ; 
and  more  recently  Dr.  Brewster  has  gone  far  toward  establish- 
ing this  doctrine.  According  to  this  eminent  optician,  (1.)  Red, 
yellow,  and  blue  light  exist  at  every  point  of  the  solar  spectrum  ; 
(2.)  As  a  certain  portion  of  red,  yellow,  and  blue,  constitute 
white  light,  the  color  of  every  point  of  the  spectrum  may  be  con- 
sidered as  consisting  of  the  predominating  color  at  any  point, 
mixed  with  white  light.  Thus  in  the  red  space,  there  is  more 
red  than  is  necessary  to  make  white  light  with  the  small  portions 
of  yellow  and  blue  which  exist  there  ;  in  the  yellow  space,  there 
is  more  yellow  than  is  necessary  to  make  white  light  with  red 
and  blue  ;  and  in  the  part  of  the  blue  space  which  appears  vio- 
let, there  is  more  red  than  yellow,  and  hence  the  excess  of  red 
forms  a  violet  with  the  blue. 

773.  The  mode  by  which  these 
three  primary  colors  produce  by  their 
combination  the  seven  colors  devel- 
oped by  the  prism,  is  exhibited  to 
the  eye  by  the  following  diagram. 
MN,  (Fig.  278,)  is  the  prismatic  Mj 
spectrum,  consisting  of  three  pri- 
mary spectra  of  the  same  length,  viz.  a  red,  a  yellow,  and  a  blue 
spectrum.  The  intensities  of  each  color  at  various  points  of  the 
spectrum,  are  represented  by  ordinates  of  different  lengths,  the 
extremities  of  which  form  the  curves  MRN,  MYN,  and  MEN, 
corresponding  to  the  three  colors  red,  yellow,  and  blue,  respec- 
tively. The  red  spectrum  has  its  maximum  intensity  at  R  ;  and 
this  intensity  may  be  represented  by  the  distance  of  the  point  R 
from  MN.  The  intensity  declines  rapidly  to  M,  and  slowly  to 
N,  at  both  of  which  points  it  vanishes.  The  yellow  spectrum  has 
its  maximum  intensity  at  Y,  the  intensity  declining  to  zero  at  M 
and  N  ;  and  the  blue  has  its  maximum  intensity  at  B,  declining 
to  nothing  at  M  and  N.  The  general  curve  which  represents 
the  total  illumination  at  any  point  will  be  outside  these  three 

GO 


522  NATURAL   PHILOSOPHY. 

curves,  and  its  ordinate  at  any  point  will  be  equal  to  the  sum  of 
three  ordinates  at  the  same  point.  Thus,  the  ordinate  of  the 
general  curve  at  the  point  Y,  will  be  equal  to  the  ordinate  of  the 
yellow  curve,  which  may  be  supposed  to  be  10  ;  added  to  that  of 
the  red  curve  which  may  be  2,  and  that  of  the  blue  which  may 
be  1.  Hence  the  general  ordinate  will  be  13.  Now  if  we  sup- 
pose that  three  parts  of  yellow,  two  of  red,  and  one  of  blue  make 
white,  we  shall  have  the  color  at  Y  equal  to  3+2+1=6  parts  of 
white  mixed  with  seven  parts  of  yellow  ;  that  is,  the  compound 
tint  at  Y  will  be  a  bright  yellow,  without  any  trace  of  red  or 
blue.  As  these  colors  all  occupy  the  same  place  in  the  spec- 
trum, they  cannot  be  separated  by  the  prism  ;  and  if  we  could 
find  a  colored  glass,  which  would  absorb  seven  parts  of  the  yel- 
low, we  should  obtain  at  the  point  Y,  a  white  light,  which  the 
prism  could  not  decompose.* 

774.  The  arguments  on  which  most  of  these  conclusions  are 
grounded,  are  derived  from  experiments  on  the  analysis  of  light 
by  absorption.  If,  (says  Dr.  Brewster,)  we  take  a  piece  of  blue 
glass  and  transmit  through  it  a  beam  of  white  light,  the  light 
will  be  of  a  fine  deep  blue.  This  blue  is  not  a  simple  homoge- 
neous color,  like  the  blue  or  indigo  of  the  spectrum,  but  is  a  mix- 
ture of  all  the  colors  of  white  light  which  the  glass  has  not  ab- 
sorbed ;  and  the  colors  which  the  glass  has  absorbed  are  those 
which  the  blue  wants  of  white  light.  In  order  to  determine  what 
these  colors  are,  let  us  transmit  through  the  blue  glass,  the  pris- 
matic spectrum  KL,  Fig.  276  ;  or,  what  is  the  same  thing,  let 


the   observer  place  his  eye  behind  the  prism  BAC,  and  look 

the  sp< 
prism,  but  with  this  remarkable  change,  that  it  will  appear  de- 


through  it ;  he  will  see  the  spectrum  on  the  other  side  of  the 


ficient  in  a  certain  number  of  its  differently  colored  rays.  A 
particular  thickness  absorbs  the  middle  of  the  red  space,  the 
whole  of  the  orange,  a  great  part  of  the  green,  a  considerable 
part  of  the  blue,  a  little  of  the  indigo,  and  a  very  little  of  the 
violet.  The  yellow  space,  which  has  not  been  much  absorbed, 
has  increased  in  breadth.  It  occupies  part  of  the  space  former- 
ly covered  by  the  orange  on  one  side,  and  part  of  the  space  for 
merly  covered  by  the  green  on  the  other.  Hence  it  follows,  that 
the  blue  glass  has  absorbed  the  red  light,  which,  when  mixed 
with  the  yellow  light,  constituted  orange,  and  has  absorbed  also 
the  blue  light,  which,  when  mixed  with  the  yellow,  constituted  a 
part  of  the  green  space  next  to  the  yellow.  We  have,  therefore, 
by  absorption,  decomposed  green  light  into  yellow  and  blue,  and 
orange  light  into  yellow  and  red ;  and  it  consequently  follows, 
that  the  orange  and  green  rays  of  the  spectrum,  though  they 
cannot  be  decomposed  by  prismatic  refraction,  can  be  decom 

*  Brewster's  Treatise  on  Optics   a.  73. 


posed  by  absorption,  «ind  actually  consist  of  two  different  colors 
possessing  the  same  degree  of  refrangibility.  Difference  of  color  is 
therefore  not  a  test  of  difference  of  refrangibility ;  and  the  con- 
clusion deduced  by  Newton  is  no  longer  admissible  as  a  gene- 
ral truth  :  "  That  to  the  same  degree  of  refrangibility  ever  be- 
longs the  same  color,  and  to  the  same  color  ever  belongs  the 
same  degree  of  refrangibility." 

By  absorbing  the  excess  of  any  color  at  any  point  of  the  spec- 
trum above  what  is  necessary  to  form  white  light,  we  may  ac- 
tually cause  white  light  to  appear  at  that  point,  and  this  white 
light  will  possess  the  remarkable  property  of  remaining  white  after 
any  number  of  refractions,  and  of  being  decomposable  only  by  ab- 
sorption. 

FIXED   LINES   IN    THE   SPECTRUM. 

775.  The  solar  spectrum,  in  its  greatest  possible  state  of  puri- 
ty and  tenuity,  when  received  on  a  white  screen,  or  when  viewed 
by  admitting  it  at  once  into  the  eye,  is  not  an  uninterrupted  line 
of  light,  red  at  one  end  and  violet  at  the  other,  and  shading  away 
by  insensible  gradations  through  every  intermediate  tint  from 
one  to  the  other,  as  Newton  conceived  it  to  be,  and  as  a  cursory 
view  shows  it.  It  is  interrupted  by  intervals  absolutely  dark ;  and 
in  those  parts  where  it  is  luminous,  the  intensity  of  the  light  is 
extremely  irregular  and  capricious,  and  apparently  subject  to  no 
law,  or  to  one  of  the  utmost  complexity.  Consequently,  if  we 
view  a  spectrum  formed  by  a  narrow  line  of  light  parallel  to 
the  refracting  edge  of  the  prism,  (which  affords  a  considerable 
breadth  of  spectrum  without  impairing  the  purity  of  the  colors, 
being,  in  fact,  an  assemblage  of  infinitely  narrower  linear  spec- 
tra arranged  side  by  side,)  instead  of  a  zone  of  equable  light  and 
graduating  colors,  it  presents  the  appearance  of  a  striped  riband, 
being  crossed  in  the  direction  of  its  breadth  by  an  infinite  mul- 
titude of  dark,  and  by  some  totally  black  bands,  distributed  ir- 
regularly throughout  its  whole  extent.  This  irregularity,  how- 
ever, is  not  a  consequence  of  any  casual  circumstances.  The 
bands  are  constantly  in  the  same  parts  of  the  spectrum,  and  pre- 
serve the  same  order  and  relations  to  each  other ;  the  same 
proportional  breadth  and  degree  of  obscurity,  whenever  and  how- 
ever they  are  examined,  provided  solar  light  be  used,  and  provided 
the  prisms  employed  be  composed  of  the  same  material ;  for  a 
difference  in  the  latter  particular,  though  it  causes  no  change  in 
the  number,  order,  or  intensity  of  the  bands,  or  their  places  in 
the  spectrum,  as  referred  to  the  several  colors  of  which  it  con- 
sists, yet  causes  a  variation  in  their  proportional  distance  from 
one  another.  By  solar  light  must  be  understood,  not  merely  the 
direct  rays  of  the  sun,  but  any  rays  which  have  the  sun  for  their 
ultimate  origin  ;  the  light  of  the  clouds,  or  sky,  for  instance  ;  of 


524 


NATURAL   PHILOSOPHY. 


the  rainbow  ;  of  the  moon,  or  of  the  planets.  All  these  lights, 
when  analyzed  by  the  prism,  are  found  deficient  in  the  identical 
rays  which  are  wanting  in  the  solar  spectrum  ;  and  the  defi- 
ciency is  marked  by  the  same  phenomenon,  viz.  by  the  occur- 
rence of  the  same  dark  bands  in  the  same  situations,  in  spectra 
formed  by  these  several  lights.  In  the  light  of  the  stars,  on  the 
other  hand,  in  electric  light,  and  in  that  of  flames,  though  simi- 
lar bands  are  observed  in  their  spectra,  yet  they  are  differently 
disposed  ;  and  the  spectrum  of  each  several  star,  and  each  flame, 
has  a  system  of  bands  peculiar  to  itself,  and  characteristic  of  its 
light,  which  it  preserves  unalterably  at  all  times,  and  under  all 
circumstances. 

776.  Pig.  279,  is  a  representation  of  the  fixed  lines  of  the 
spectrum,  according  to  Fraunhofer,*  the  small  bands  observed 
by  him  (more  than  five  hundred  in  number)  being  omitted.  Of 
these  fixed  lines,  he  selected  seven,  (those  marked  B,  C,  D,  E,  F, 
G,  H,)  as  terms  of  comparison,  or  as  standard  points  of  reference 
in  the  spectrum,  on  account  of  their  distinctness,  and  the  facility 
with  which  they  may  be  recognised.  The  definiteness  of  these 
lines,  and  their  fixed  position  with  respect  to  the  colors  of  the 
spectrum, — in  other  words,  the  precision  of  the  limits  of  those 

Fig.  279. 
Violet  Indigo.         Blue.         Green.    Yellow.  Orange.    Red. 


H  G  F  D  CB     A 

degrees  of  refrangibility  which  belong  to  the  deficient  rays  of  so 
lar  light, — renders  them  invaluable  in  optical  inquiries,  and  ena- 
bles us  to  give  a  precision  hitherto  unheard  of  to  optical  meas- 
urements, and  to  place  the  determination  of  the  refractive  pow- 
ers of  media  on  the  several  rays  almost  on  the  same  footing,  in 
respect  to  exactness,  with  astronomical  observations.! 


NATURE    OP   LIGHT. 

777.  The  phenomena  of  light  may  be  explained,  either  on  the  sup- 
position that  light  is  a  material  fluid  of  extreme  subtilty,  or  that  it 
is  produced  by  the  undulations  of  an  independent  medium,  set  in 
motion  by  the  luminous  body. 

Opticians  of  great  eminence,  as  Descartes,  Huygens,  Euler, 
and  Young,  have  held  the  opinion,  that  light  does  not  consist  of 


*  A  celebrated  German  optician,  recently  deceased.  t  Herschel  on  Light. 


OPTICS.  525 

actual  emanations  of  material  particles  from  the  luminous  body, 
but  that  such  a  body  has  merely  the  property  of  communicating 
a  series  of  vibrations  to  a  peculiar  fluid  that  is  diffused  through- 
out the  universe,  which  vibrations  form  the  communication  be 
tween  the  luminous  body  and  the  eye.  The  medium  is  conceived 
to  be  of  extreme  tenuity  and  elasticity  ;  such,  indeed,  that  though 
filling  all  space,  it  shall  offer  no  appreciable  resistance  to  the 
motions  of  the  planets  and  comets,  capable  of  disturbing  them 
in  their  orbits.  It  is,  moreover,  imagined  to  penetrate  all  bodies ; 
but  in  their  interior,  to  exist  in  a  different  state  of  intensity  and 
elasticity  from  those  which  belong  to  it  in  a  disengaged  state, 
and  hence  the  refraction  and  reflexion  of  light.  Newton,  how- 
ever, and  with  him  the  greater  number  of  opticians,  have  held, 
that  light  consists  of  actual  particles  of  matter  sent  off  from  lu- 
minous objects  to  the  eye.  In  the  former  case,  the  fluid  is  only 
the  medium  of  light,  as  air  is  the  medium  of  sound  ;  the  vibra- 
tions of  the  medium  following  each  other,  as  wave  follows  wave, 
with  incredible  swiftness,  and  thus  conveying  the  impression 
from  the  radiant  to  the  eye :  in  the  latter  case,  the  motion  is 
simply  that  of  a  chain  of  particles  moving  in  right  lines  with  the 
same  astonishing  velocity.  Thus,  when  the  sun  rises,  it  either 
sends  forth  luminous  particles,  which,  entering  the  eye,  occasion 
the  sensation  of  vision ;  or  puts  in  motion  the  peculiar  fluid 
which  is  the  medium  of  light,  which  motion  is  propagated  from 
wave  to  wave  till  it  reaches  the  eye. 

778.  It  forms  a  strong  objection  against  the  hypothesis  of  un- 
dulations, that  the  motions  of  light  are  confined  to  right  lines,  a 
condition  not  essential  to  this  species  of  motion  ;  while  it  is  a 
strong  argument  in  favor  of  the  materiality  of  light,  that  it  ex- 
hibits the  property  of  attraction,  one  of  the  most  characteristic 
properties  of  matter.  The  motion  is  conformable  to  the  laws 
which  regulate  the  motions  of  small  bodies  under  the  same  cir- 
cumstances. Thus,  when  it  meets  with  no  impediment,  it  moves 
uniformly  forward  in  right  lines ;  it  is  affected  by  passing  into 
mediums  of  different  densities,  in  a  manner  correspondent  to  the 
law  of  the  mutual  gravitation  of  matter,  being  attracted  from 
rarer  toward  denser  bodies ;  and  finally,  it  produces  certain 
chemical  changes  in  bodies  which  belong  to  none  but  a  material 
agent.  The  rays  of  light,  also,  by  passing  through  certain  me- 
dia, undergo  a  change,  to  be  described  hereafter  under  the  head 
of  polarization,  in  which  the  opposite  sides  of  a  ray  appear  to  be 
endowed  with  different  properties,  a  fact  which  accords  ill  with 
the  idea  of  undulations,  though  it  is  quite  consistent  with  the 
doctrine  of  the  materiality  of  light.  The  latter  hypothesis, 
moreover,  has  the  advantage  of  leading  the  student  to  a  more 
ready  apprehension  of  the  nature  of  optical  phenomena.  Still, 
the  object  of  this  science  is  not  so  much  to  ascertain  the  nature 


526  NATURAL   PHILOSOPHY. 

of  the  agent  on  which  the  phenomena  of  light  depend,  as  it  is 
to  study  those  phenomena  themselves,  and  to  classify  them  under 
general  laws,  which  may  be  applied  to  the  construction  of  optical 
instruments,  and  to  the  interpretation  of  nature. 

779.  To  the  doctrine  of  the  materiality  of  light,  it  has  been 
objected,  first,  that  material  particles,  endued  with  such  immense 
velocity,  would  have  a  momentum  which  nothing  could  resist, 
much  less  so  delicate  an  organ  as  the  eye  ;  secondly,  that  were 
the  rays  material,  so  prodigious  is  their  number  scattered 
throughout  the  universe,  they  would  interfere  with  one  another , 
and,  thirdly,  that  the  sun  and  stars  would  waste  away  and  grow 
dim,  by  such  a  constant  and  profuse  expenditure  of  matter.  But 
these  objections  severally  admit  of  a  satisfactory  reply.  In  the 
first  place,  the  momentum  of  a  ray  of  light  may  still  be  incon- 
siderable, if  the  quantity  of  matter  is  small  in  the  same  propor- 
tion as  the  velocity  is  great.  Though  such  an  attenuation  of 
matter  is  amazing,  yet  it  is  not  incredible,  but  perfectly  consist- 
ent with  the  known  analogies  of  nature.  In  the  second  place, 
notwithstanding  the  universal  diffusion  of  light,  no  interference 
of  its  particles  is  necessary,  for  it  is  not  essential  to  the  purpose 
of  vision,  that  a  ray  should  consist  of  contiguous  particles  of 
light.  It  is  found,  that  the  sensation  continues  for  some  time 
after  the  luminous  object  is  removed,  during  an  interval  suffi- 
cient for  light  to  pass  through  twenty-two  thousand  miles  ;  con- 
sequently, particles  no  nearer  to  each  other  than  this  distance, 
would  be  competent  to  maintain  uninterrupted  vision.  Thus,  an 
ignited  stick,  whirled  in  the  air,  exhibits  a  ring  of  light,  because 
the  sensation  continues  for  a  longer  time  than  the  illuminated 
point  occupies  in  passing  round  the  circle.  In  the  third  place, 
the  small  danger  of  waste  sustained  by  the  sun  in  consequence 
of  the  light  which  it  dispenses,  may  be  inferred  from  the  follow- 
ing remarks  of  Dr.  Priestley.  After  relating  an  experiment,  in 
which  the  light  of  the  sun  collected  during  one  second,  by  a  con- 
cave reflector  of  four  square  feet,  and  thrown  on  the  arm  of  a 
delicate  balance,  indicated  a  weight  not  exceeding  the  1200  mil- 
lionth part  of  a  grain,  the  Doctor  adds  :  "  Now  the  light  in  the 
above  experiment  was  collected  from  a  surface  of  four  square 
feet,  which  reflecting  only  about  half  what  falls  upon  it,  the 
quantity  of  matter  contained  in  the  rays  of  the  sun  incident  upon 
a  square  foot  and  a  half  of  surface  in  one  second  of  time,  ought 
to  be  no  more  than  the  1200  millionth  part  of  a  grain;  But  the 
density  of  light  at  the  surface  of  the  sun  is  greater  than  at  the 
earth  in  the  proportion  of  45000  to  1  ;  there  ought,  therefore,  to 
issue  from  one  square  foot  of  the  sun's  surface,  in  one  second  of 
time,  in  order  to  supply  the  waste  by  light,  one  forty  thousandth 
part  of  a  grain  of  matter, — that  is,  a  little  more  than  two  grains 


OPTICS.  527 

in  a  day,  or  about  4752000  grains,  which  is  about  670  pounds, 
avoirdupois,  in  six  thousand  years."* 

The  more  recent  and  refined  discoveries  in  optics  have  favored 
the  theory  of  undulations,  and  the  argument  derived  from  the 
authority  of  great  names  is  now  decidedly  on  the  side  of  this 
theory,  and  against  that  of  emissions.! 


CHAPTER  VI. 

OF  COLORS  IN  NATURAL  OBJECTS. 

780.  THE  knowledge  of  the  composition  of  light,  and  of  the 
properties  of  the  solar  spectrum,  naturally  led  to  an  inquiry  into 
the  subject  of  colors,  as  exhibited  in  the  phenomena  of  nature. 
The  bright  tints  of  the  rainbow — the  splendid  hues  sometimes 
exhibited  by  thin  plates,  as  soap  bubbles — and  finally,  the  diver- 
sified colors  of  objects  in  all  the  kingdoms  of  nature,  remained  to 
be  accounted  for.     We  propose  now  to  inquire  how  far  this  ob- 
ject has  been  effected. 

THE    RAINBOW. 

781.  The  rainbow,  one  of  the  most  striking  and  magnificent 
of  the  phenomena  of  nature,  was  long  ago  supposed  to  be  owing 
to  some  modification  which  the  light  of  the  sun  undergoes  in 
passing  into  drops  of  rain  ;  but  the  complete  development  of  the 
causes  on  which  it  depends,  was  reserved  for  the  genius  of  New- 
ton, and  naturally  followed  in  the  train  of  those  discoveries  which 
he  made  upon  the  prismatic  spectrum. 

The  rainbow,  when  exhibited  in  its  more  perfect  forms,  con- 
sists of  two  arches,  usually  seen  in  the  east  during  a  shower  of 
rain,  while  the  sun  is  shining  in  the  west.  These  arches  are  de- 
nominated the  outer  and  the  inner  bow,  of  which  the  inner  bow 
is  the  brighter,  but  the  outer  bow  is  of  larger  dimensions  every 
way.  The  succession  of  colors  in  the  one  is  directly  opposite  to 
that  of  the  other. 

782.  Drops  of  rain,  though  small,  are  large  in  comparison 
with  the  minuteness  of  rays  of  light,  and  are  to  be  regarded  as 
spheres  of  water,  exerting  the  powers  of  refraction  and  reflexion 
in  the  same  manner  as  large  globes  of  water  would  do.     It  was, 

*  Priestley,  Hist.  Light  and  Colors. 

t  See  an  excellent  view  of  the  doctrine  of  Undulations  in  Pouillet's  Traite"  de  Phy- 
eique,  II.  263. 


OJJO  NATURAL   PHILOSOPHY. 

in  fact,  by  investigating  the  manner  in  which  globular  glass  ves- 
sels filled  with  water  modify  the  solar  rays,  that  the  first  hints 
were  obtained  respecting  the  cause  of  the  rainbow.  In  the  year 
1611,  Antonio  de  Dominis  made  a  considerable  advance  toward 
the  theory  of  the  rainbow,  by  suspending  a  glass  globe  in  the 
sun's  light,  when  he  found,  that  while  he  stood  with  his  back  to 
the  sun,  the  colors  of  the  rainbow  were  reflected  to  his  eye  in 
succession  by  the  globe,  as  it  was  moved  higher  or  lower. 
Let  us,  therefore,  in  the  first  place,  fol-  Fig.  280. 

low  the  course  of  a  ray  of  light  through  a 
globule  of  water.  Let  SA  (Fig.  280)  be  a 
small  beam  of  light  from  the  sun,  falling 
upon  the  surface  of  a  globule  of  water  at 
A.  Agreeably  to  what  is  known  of  the 
laws  of  light  in  passing  out  of  one  trans- 
parent medium  into  another,  a  portion  of 
the  rays  would  be  reflected  at  A,  and  an- 
other portion  would  pass  into  the  drop  and 
be  refracted  to  the  further  surface  at  B.  The  same  effect  would 
recur  here,  and  also  at  D  and  at  F  ;  and  were  the  eye  situated 
in  either  of  the  lines  BC,  DE,  or  FG,  it  would  perceive  the  pris- 
matic colors,  because  some  of  the  rays  which  composed  the 
beam  of  light  that  reached  the  eye,  would  be  refracted  more 
than  others,  and  thus  the  different  colors  would  be  made  to  ap- 
pear. Or  if  a  screen  were  so  placed  as  to  receive  these  trans 
mitted  rays,  a  faint  spectrum  would  be  formed  upon  it.  Such  a 
progress  of  a  beam  of  light  admitted  through  the  window  shut- 
ter, and  made  to  fall  on  a  globular  vessel  of  water,  may  be  actu- 
ally rendered  visible  by  experiment.* 

783.  It  may  be  remarked,  that  but  a  comparatively  small  part 
of  the  solar  rays  that  shine  upon  a  drop  of  water,  are  required 
in  order  to  produce  the  mild  light  of  the  rainbow,  aided  as  its 
light  is  by  the  dark  ground  or  cloud  on  which  it  is  usually  pro- 
jected ;  yet  where  the  number  of  rays  that  enter  the  eye  is  di- 
minished beyond  a  certain  limit,  the  light  becomes  too  feeble  for 
distinct  vision.  It  will  also  be  observed,  that  a  considerable 
portion  of  light  is  lost  at  each  successive  reflexion  that  takes 
place  within  the  drop,  so  that  a  certain  beam  of  light,  conveyed 
to  the  eye  after  two  reflexions,  will  be  much  more  feeble  than 
the  same  beam  after  one  reflexion.  Indeed,  so  much  of  the  sun's 
light  is  dissipated  at  the  first  point  of  reflexion  from  the  interior 
surface,  added  to  what  is  transmitted  at  the  same  point,  and  of 
course  never  reaches  the  eye  of  the  spectator,  that,  were  it  not 
for  a  great  accumulation  which  the  sun's  rays  undergo  at  a  par 
ticular  point  in  the  drop,  whence  the  light  is  reflected  and  con- 

*  Biot. 


CPTICS.  529 

veyed  to  the  eye,  the  phenomena  of  the  rainbow  would  not  oc- 
cur. The  manner  in  which  this  accumulation  is  effected,  is  now 
to  be  explained. 

784.  Let  fzpq  (Fig.  281)  be  the  section  of  a  drop  of  ra,in,fp  a 
diameter,  ab,  cd,  &c.  parallel  rays  of  the  sun's  light,  falling  upon 
the  drop.  Now  yf,  a  ray  coinciding  with  the  diameter,  would 

Fig.  281. 


suffer  no  refraction  ;  and  ab,  a  ray  near  to  yf,  would  suffer  only 
a  very  small  inclination  toward  the  radius,  so  as  to  meet  the  re- 
moter surface  of  the  drop  very  near  to  p  ;  but  the  rays  which  lie 
further  from  yf,  being  inclined  toward  the  radius  in  a  greater 
angle,  would  be  more  and  more  refracted  as  they  were  further 
removed  from  the  diameter.  The  consequence  would  be,  that 
after  passing  a  certain  limit,  the  rays  that  lay  above  that  limit 
would  cross  those  which  lay  below  it,  and  meet  the  further  sur- 
face somewhere  between  the  diameter  and  the  ray  which  passed 
through  the  said  limit ;  that  is,  all  the  rays  falling  on  the  quad- 
rant fz,  would  meet  the  circumference  within  the  arc  Ap.  Bat 
when  a  quantity  is  approaching  its  limit,  or  is  beginning  to  devi- 
ate from  it,  its  variations  are  nearly  insensible.  Thus,  when  the 
sun  is  at  the  tropics,  being  the  limits  to  which  he  departs-  from 
the  equator,  he  appears  for  some  time  to  remain  at  the  same 
point.  In  the  same  manner,  a  great  number  of  the  rays  which 
lie  contiguous  to  cd,  on  both  sides  of  it,  will  meet  in  very  nearly 
the  same  point  on  the  concave  surface  of  the  drop  at  km.  Con- 
sequently, a  greater  number  of  rays  will  be  reflected  from  that 
point  than  from  any  other  in  the  arc.  Now  were  these  rays  to 
return  in  the  same  lines,  they  would  emerge  parallel  in  the  lines 
cd,  no  ;  but  if,  instead  of  returning  back  in  the  quadrant  fz,  they 
are  reflected  on  the  other  side  of  the  radius,  they  may  meet  the 

67 


530 


NATURAL   PHILOSOPHY. 


curve  at  the  same  angle  in  the  quadrant  fq,  and  emerge  paral- 
lel, coming  to  the  eye  at  s.  Hence  it  appears,  that  there  is  a 
Particular  point  in  a  drop  of  rain,  where  the  ra"ys  of  the  sun's 
ght  seem  to  accumulate,  and  are  therefore  peculiarly  fitted  to 
make  an  impression  on  the  organ  of  vision.  It  is  found  by  cal- 
culation, that  the  angle  which  the  incident  and  emergent  rays, 
in  such  cases,  make  with  each  other,  is,  for  the  red  rays  42°  2', 
and  for  the  violet  rays  40°  17'.  These  are  the  angles  when  the 
rays  emerge  after  two  refractions  and  one  reflexion  :  in  the  case 
of  two  refractions  and  two  reflexions,  the  angles  are,  for  the  red 
rays  50°  59',  and  for  the  violet  54°  9'. 

785.  Let  us  next  consider  what  must  be  the  position  of  the 
spectator,  in  order  that  his  eye  may  receive  the  emergent  rays 
which  make  the  foregoing  angle  with  the  incident  rays,  and 
which,  of  course,  are  those  which  cause  the  phenomena  of  the 
rainbow. 

The  spectator  must  stand  with  his  back  to  the  sun,  and  a  line 
drawn  from  the  sun  toward  the  bow  so  as  to  pass  through  his  eye, 
will  make  the  same  angle  with  the  emergent  rays  that  these  make 
with  the  incident  rays.  Thus,  let  AB  (Fig.  282.)  be  the  incident, 
and  GI  the  emergent  ray,  and  let  the  angle  which  these  two  rays 
make  with  each  other  be  AKI ;  and  let  IT  be  a  ray  passing  from 
the  sun  toward  the  bow  through  the  eye  of  the  spectator ;  then, 
(since  the  rays  of  the  sun  may  be  regarded  as  parallel,)  AB  and 
IT  are  parallel,  and  the  alternate  angles  AKI  and  KIT,  equal. 
Fig.  282. 


But  AKI  is  the  angle  made  by  the  incident  and  emergent  rays, 
and  KIT  the  angle  made  by  the  emergent  ray,  and  a  line  drawn 
from  the  sun  toward  the  bow  through  the  eye  of  the  spectator. 

786.  When  the  sun  shines  upon  the  drops  of  rain  as  they  are 
falling,  the  rays  which  come  from  those  drops  to  the  eye  of  the  spec- 
tator after  ONE  REFLEXION  AND  TWO  REFRACTIONS,  produce  the  inner- 
most or  primary  rainbow ;  and  those  drops  which  come  to  the  eye 


OPTICS.  531 

after  TWO  REFLEXIONS  AND  TWO  REFRACTIONS,  produce  the  outermost 
or  superior  rainbow. 

Let  SOC*  (Fig.  283)  be  a  straight  line  passing  from  the  center 
of  the  sun  through  the  eye  of  the  spectator  at  O,  toward  the  bow, 
and  let  SR,  SV,  be  incident  rays,  which,  after  one  reflexion  and 
two  refractions,  are  conveyed  to  the  eye  at  O,  making,  (Art.  785,) 
with  SOC,  angles  equal  to  those  formed  by  the  incident  and 
emergent  rays.  If  O  V  makes  with  SOC  an  angle  of  40°  17',  and 
be  conceived  to  revolve  around  OC,  describing  the  surface  of  a 
cone,  all  the  drops  of  rain  on  this  surface  will  be  precisely  in  the 
situation  necessary  in  order  that  the  Fig.  283. 
violet  rays,  after  two  refractions  and 
one  reflexion,  may  emerge  parallel  and 
arrive  at  the  eye  in  O,  and  this  will  not 
take  place  in  the  same  manner  on  any 
other  part  of  the  cloud ;  so  that,  by 
means  of  this  species  of  rays,  the  spec- 
tator will  see  on  the  cloud  a  violet  col- 
ored arc,  of  which  OC  will  be  the  axis, 

and  C  the  center.     He  will,  besides,        so  C 

see  also  an  infinity  of  other  concentric  arcs  exterior  to  the  vio- 
let, each  one  of  which  will  be  made  up  of  a  single  species  of 
rays;  and  according  as  these  rays  are  less  refrangible,  their  areas 
will  be  of  greater  diameter,  so  that  the  largest,  composed  of  the 
extreme  red,  will  subtend  an  angle  ROC  of  42°  2'.  Therefore 
the  whole  width  of  the  colored  bow  will  be  42°  2'— 40°  17',  or 
1°  45',  the  red  being  on  the  outside  and  the  violet  within. 

The  contrary  order  of  colors  will  result  from  two  reflexions  and 
two  refractions.  Let  SV,  SR',  be  the  incident  rays,  which,  after 
two  reflexions  and  two  refractions,  converge  to  the  eye  at  O, 
making  (Art.  785,)  with  SOC,  angles  equal  to  those  formed  by 
the  incident  and  emergent  rays,  namely,  50°  59'  and  54°  9',  and 
the  lines  R'O  and  V'O,  as  before,  be  conceived  to  revolve  around 
SOC,  they  will  severally  meet  with  all  the  drops,  which  having 
twice  refracted  and  twice  reflected  the  extreme  red  and  violet 
rays,  can  transmit  them  to  the  eye  parallel  to  each  other.  Be- 
tween these  two  arcs,  there  will  be  others  exhibiting  all  the  in- 
termediate prismatic  colors ;  and  the  whole  together  will  form 
a  second  bow,  whose  breadth  will  be  54°  9'  — 50°  59',  or  3°  10*. 

787.  The  rays,  therefore,  which  come  from  all  the  drops  which 
make  an  angle  of  42Q  2'  with  a  line  passing  from  the  sun  through 
the  eye  (which  may  be  called  the  axis  of  vision,)  appear  red ; 
and  it  is  obvious,  that  a  collection  of  rays  drawn  all  around  this 
axis  from  the  eye  to  drops  thus  situated,  would  form  a  cone,  of 

*  It  will  be  observed,  that  the.  line  SOC  w  at  right  angles  to  the  plane  of  the  sur- 
face ;  that  is,  to  the  plane  of  the  bows 


532  NATURAL   PHILOSOPHY. 

which  the  drops  themselves  would  constitute  the  base,  and  of  course 
would  form  a  circle.  The  same  is  true  of  all  the  other  colors 
which  emerge  from  drops  at  angles  which  are  different  for  dif- 
ferent colors,  but  constant  for  the  same  color.  Hence,  the  line 
which  passes  from  the  sun  through  the  eye  of  the  spectator,  passes 
also  to  the  center  of  tht  bow.  or  is  the  axis  of  the  cone  of  which  the 
bow  itself  is  the  base.  If  the  sun  is  on  the  horizon,  this  axis 
becomes  a  horizontal  line  ;  consequently,  the  center  of  the  arch 
rests  on  the  opposite  horizon,  and  the  bow  is  a  semicircle,  of 
which  the  highest  point  has  an  altitude  above  the  horizon  of 
42°  2'.  If  the  sun  is  at  this  altitude  of  42°  2'  above  the  horizon, 
then  the  center  of  the  bow  will  have  the  same  depression  be- 
low the  opposite  horizon,  and  the  circumference,  at  its  highest 
point,  will  just  reach  that  horizon.  When  the  sun  is  between 
these  two  points,  the  elevation  of  the  bow  will  be  the  difference 
between  the  altitude  of  the  sun  and  the  foregoing  angle. 

788.  When  the  spectator  is  on  an  eminence,  as  a  high  moun- 
tain, he  may  see  more  than  half  the  bow,  when  the  sun  is  near 
setting  ;  for  the  axis  will  in  that  case  pass  to  a  point  above  the 
opposite  horizon.      Travellers  who  have  ascended  very  high 
mountains,  have  occasionally  observed  their  shadows  projected 
on  the  clouds  below,  with  their  heads  encircled  with  rainbows.* 
In  this  case,  the  axis  passes  to  a  point  above  the  opposite  hori- 
zon equal  to  or  greater  than  the  semi-diameter  of  the  bow,  so 
that  the  whole  of  the  circumference  comes  into  view ;  and  the 
eye  of  the  spectator  being  in  the  axis,  the  entire  bow  is  projected 
around  that  as  a  center,  upon  the  surface  of  the  clouds. 

COLORS  OF  BODIES. 

789.  According  to  the  Newtonian  theory,  the  color  of  a  body 
depends  on  the  kind  of  light  which  it  rejlects.     A  great  number  of 
bodies  are  fitted  to  reflect  at  once  several  kinds  of  rays,  and  con- 
sequently appear  under  mixed  colors.     It  may  even  happen  that 
of  two  bodies  which  should  be  green,  for  example,  one  may  re- 
flect the  pure  prismatic  green,  and  the  other  the  green  which  ari- 
ses from  the  mixture  of  yellow  and  blue.     This  quality  of  selec- 
tion, as  it  were,  in  bodies,  which  varies  to  infinity,  occasions  the 
different  kinds  of  rays  to  unite  in  every  possible  manner  and  eve* 
ry  possible  proportion ;  and  hence  the  inexhaustible  variety  of 
shades  which  nature,  as  in  sport,  has  diffused  over  the  surfaces  of 
different  bodies. 

When  a  body  absorbs  nearly  all  the  light  that  reaches  it,  that 
body  appears  black ;  it  transmits  to  the  eye  so  few  reflected 

*  Amer.  Jour,  of  Science,  xii,  172. — Malte"  L  Tin's  Universal  Geography,  vol.  1,  p. 


OPTICS.  533 

rays,  that  it  is  scarcely  perceptible  in  itself,  and  its  presence  and 
form  make  no  impression  on  us,  unless  as  it  interrupts,  in  a  man- 
ner, the  brightness  of  the  surrounding  space. 

But  for  a  body  to  reflect  one  kind  of  ray  rather  than  any  oth- 
er kind,  there  must  be  something  in  that  body  which  determines 
the  preference.  In  what  then  does  a  red  body  differ  in  this  re- 
spect from  a  yellow,  a  green,  or  a  violet  one  ?  Various  attempts 
have  been  made,  and  on  various  hypotheses,  to  resolve  this  ques- 
tion. Newton,  who  entered  on  this  subject  with  great  earnest- 
ness, has  here  most  successfully  interrogated  nature  by  a  series 
of  experiments,  of  which  we  shall  give  the  results. 

790.  Having  taken  two  glasses  of  a  telescope,  the  one  plano- 
convex, the  other  slightly  convex  on  both  sides,  he  placed  one  of 
the  faces  of  this  upon  the  plane  face  of  the  former,  and  pressed 
the  two  glasses  at  first  gently,  and  then  by  degrees  more  closely 
against  one  another.  The  effect  of  this  gradual  pressure  was, 
an  appearance  of  colored  circles  in  the  plate  of  air  between  the 
glasses,  which  circles  had  the  point  of  contact  for  a  common 
center,  and  which  increased  in  number  as  the  pressure  was  in- 
creased, in  such  a  manner  that  the  circle  which  appeared  last 
always  was  nearest  the  point  of  contact,  and  on  a  still  further 
pressure  extended  its  circumference  to  form  a  kind  of  ring  round 
a  new  circle  that  arose  near  its  middle. 

The  pressure  having  been  carried  to  a  certain  term,  Newton 
stopped  and  observed  as  follows :  At  the  point  of  contact  was  a 
black  spot  that  was  encompassed  by  several  series  of  colors,  ar- 
ranged from  the  center  outward  in  the  following  order : 

First  series,  blue,  white,  yellow,  red. 

Second,  violet,  blue,  green,  yellow,  red. 

Third,  purple,  blue,  green,  yellow,  red. 

Fourth,  green  and  red. 

Fifth,  greenish  blue  and  red. 

Sixth,  greenish  blue  and  pale  red. 

Seventh,  greenish  blue  and  reddish  white. 

Beyond  this  number  the  tints  were  regularly  paler  until  the 
color  became  white. 

The  reason  why  these  successive  colors  were  arranged  in  rings, 
having  the  point  of  contact  of  the  two  lenses  for  their  common 
center,  is  obvious,  since  each  color  was  developed  at  a  certain 
thickness,  and  the  points  of  equal  thickness  being  equidistant 
from  the  center,  they  would,  of  course,  be  arranged  in  the  cir- 
cumference of  a  circle. 

Newton  measured  the  diameters  of  the  annular  bands  formed 
of  these  different  colors,  by  taking  the  points  where  they  had 
most  lustre  ;  and  he  found  that  the  squares  of  those  diameters 
were  to  one  another  as  the  terms  of  the  ascending  progression 
1.  3,  5,  7,  9,  11,  &c. :  from  which  it  results,  that  the  intervals  be- 


534  NATURAL   PHILOSOPHY. 

tween  the  two  glasses,  relatively  to  the  corresponding  points,  fol- 
lowed the  same  progression. 

For,  let  nam  (Fig.  284,)  be  a  di- 
ameter taken  on  the  surface  of  the 
plane  glass,  and  agf  a  section  of 
the  sphere  to  which  that  part  of  the 
double  convex  lens  that  turns  to- 
ward a,  belongs.  Let  also  ab,  ad, 
be  the  semi-diameters  of  the  two 
rings  at  the  points  where  the  colors 
are  most  vivid.  Having  drawn  be, 
dg,  parallel  to  the  diameter  af,  and 
eh,gi,  parallel  to  an,  we  shall  have 

(eh)3:  (gi)'  ::  ahxhf :  aixif.         n          db       a  m 

But  the  distances  between  the  two  lenses  being  exceedingly 
small  in  comparison  with  the  diameter  af,  hf  and  if  may  be 
taken  as  equal  to  af,  whence,  by  substitution, 

(eh)3  :  (gi)s :  :  ah xaf :  aixaf :  :  ah  :  ai : :  be  :  dg. 

791.  From  these  proportions,  it  was  merely  necessary  to  ascer- 
tain the  absolute  length  of  a  single  diameter,  to  know  the  lengths 
of  all  the  others,  as  well  as  the  different  thicknesses  of  the 
plates  of  air  at  the  points  where  the  different  colors  were  seen. 
Newton  drew  up  a  table  of  these  degrees  of  thickness,  assigning 
to  each  color  that  degree  at  which  it  was  developed.  For  exam- 
ple, the  most  intense  blue  makes  its  appearance  at  a  thickness 
equal  to  the  two  millionth  part  of  an  inch,  the  visual  ray  being 
supposed  to  come  to  the  eye  perpendicularly  to  the  two  glasses. 
As  the  visual  rays  deviate'  from  a  perpendicular,  the  breadth  of 
the  rings  increases,  the  same  color  requiring  a  greater  thickness 
to  produce  it.  Among  other  results  obtained  from  these  experi- 
ments, were  the  following : 

Air,  at  and  below  a  thickness  of  half  a  millionth  of  an  inch, 
ceases  to  reflect  light.  At  and  above  a  thickness  of  seventy-two 
millionth^  of  an  inch,  it  reflects  white ;  that  is,  all  the  rays  of  the 
spectrum.  Between  these  two  limits,  it  reflects  the  various  or- 
ders of  colors  contained  in  the  table. 

Water,  at  and  below  a  thickness  of  three  eighths  of  a  millionth 
of  an  inch,  ceases  to  reflect  light.  At  and  above  fifty-eight 
millionths  of  an  inch,  it  reflects  white ;  and  between  these  two 
limits,  it  reflects  the  orders  of  the  colors  contained  in  the  table. 

Glass,  at  and  below  one  third  of  a  millionth  of  an  inch,  ceases 
to  reflect  light.  At  and  above  a  thickness  of  the  fifty  millionth 
of  an  inch,  it  reflects  white  ;  and  between  these  limits,  it  reflects 
the  orders  of  colors  contained  in  the  table. 

Newton  also,  having  measured  the  diameters  of  the  rings  at 
ths  intermediate  places  where  the  colors  were  obscure,  found  that 
their  squares  were  to  one  another  as  the  even  numbers  2,  4,  6, 


OPTICS.  535 

8,  10,  12,  &c. ;  and  hence  the  intervals  between  the  glasses,  at 
the  corresponding  points,  observed  a  similar  progression. 

792.  Such  were  the  phenomena  which  the  glasses  presented 
as  seen  by  reflexion  ;  but  on  looking  through  them  to  observe  the 
effect  of  refracted  light,  other  series  of  colors  took  the  place  of  the 
preceding  ones.     The  central  spot,  which  was  before  black,  now 
became  white,  and  the  order  of  colors,  relatively  to  the  different 
series,  was  this : 

1.  Yellowish- red,  black,  violet,  blue. 

2.  White,  yellow,  red,  violet,  blue. 

3.  Green,  yellow,  red,  bluish-green. 

4.  5,  6.  Red,  bluish-green. 

By  comparing  these  colors  seen  by  transmitted,  with  those  seen 
by  reflected  light,  it  is  observable,  that  the  white  answers  to 
black,  the  red  to  blue,  the  yellow  to  violet,  the  green  to  a  mixture 
of  red  and  violet ;  that  is,  the  part  that  appeared  black  on  sim- 
ply looking  at  the  glasses,  became  white  when  the  observer  look- 
ed through  them,  and  so  of  the  other  colors.  But  the  tints  pro- 
duced by  transmitted  light  were  feeble  and  languishing,  unless 
the  visual  ray  was  extremely  oblique,  in  which  case  they  were 
sufficiently  vivid  and  brilliant. 

793.  Newton  substituted  water  for  air  between  the  two  glasses, 
and  the  colors  instantly  became  fainter,  and  the  rings  contracted ; 
that  is,  the  ring  of  a  particular  color  had  its  circumference  near- 
er the  center  than  when  that  color  was  reflected  by  the  plate  of 
air.     The  diameters  of  the  corresponding  rings  were  to  one  anoth- 
er nearly  as  7  to  8,  and  consequently  their  squares  were  as  49  to 
64 ;  whence  it  follows,  that  the  different  thicknesses  of  the  fluids 
at  the  places  where  the  rings  appeared,  were  nearly  as  3  to  4  ; 
that  is,  in  the  ratio  of  the  sine  of  incidence  to  the  sine  of  refrac- 
tion (Art.  749,)  when  the  light  passes  from  water  into  air.     New- 
ton imagined  that  this  result  might  be  extended  to  all  kinds  of 
mediums,  and  he  therefore  deduced  from  it  this  general  law : 
that  where  a  medium  more  or  less  dense  than  water  is  impress- 
ed between  two  glasses,  the  interval  between  the  glasses  at  the 
place  where  any  particular  color  is  perceived,  is  to  the  interval 
which  gives  that  color  by  means  of  air,  as  the  sines  which  meas- 
ure the  refraction  at  the  passage  from  the  same  medium  into 
air.     This  rule  may  be  equally  applied  to  a  thin  plate,  detached 
from  any  kind  of  body,  the  thickness  of  which  we  would  deter- 
mine by  the  tone  of  its  color.* 

794.  The  phenomena  of  the  rings  being  reduced  to  laws  ex- 
tremely exact  and  well  adapted  to  calculation,  Newton  reduced 

*  Haiiy  Nat.  Philosophy,  Sects.  711—720. 


536  NATURAL   PHILOSOPHY. 

them  all  to  a  still  simpler  expression,  making  them  depend  on  a 
physical  property  which  he  attributed  to  light,  and  of  which  he 
defined  all  the  particulars  conformably  to  their  laws.  Consider- 
ing, light  as  a  matter  composed  of  small  molecules  emitted  by 
luminous  bodies  with  very  great  velocities,  he  concluded,  that 
since  they  were  reflected  from  a  lamina  of  air,  at  the  several 
thicknesses  corresponding  to  the  numbers  1,  3,  5,  7,  &c.,  and 
transmitted  at  the  intermediate  thicknesses  0,  2,  4,  6,  &c.,  the 
molecules  must  have  some  peculiar  modification  of  a  periodical 
nature,  such  as  to  incline  them  alternately  to  be  reflected  and 
refracted  after  passing  through  certain  spaces.  Newton  charac- 
terized this  tendency  to  alternate  reflexion  and  transmission,  and 
designated  the  two  states  by  the  phrases  jfe  of  easy  reflexion  and 
Jits  of  easy  transmission.* 

795.  Having  defined  completely  all  the  characters  of  these  fits, 
or  periodical  returns  of  states  favorable  to  reflexion  and  transmission, 
Newton  employed  them  as  a  simple  property,  not  only  to  unite 
under  one  point  of  view  the  phenomena  of  the  colors  produced 
by  thin  plates,  but  also  to  foresee  and  to  calculate  beforehand. 
both  as  to  their  general  tenor,  and  their  minutest  details,  a  crowd 
of  analogous  phenomena,  observed  to  attend  reflexion  in  thick 
plates,  which,  in  fact,  exceeded  by  as  much  as  twenty  or  thirty 
thousand  times  those  on  which  the  calculations  had  been  found- 
ed ;  moreover,  applying  the  same  reasoning  to  the  integrant  par- 
ticles of  material  substances,  which  all  chemical  and  physical 
phenomena  show  to  be  very  minute,  and  to  be  separated  even 
in  the  most  solid  bodies,  by  spaces  immense  in  comparison  of 
their  absolute  dimensions,  he  was  able  to  deduce  naturally  from 
the  same  principles  the  theory  of  the  different  colors  they  present 
to  us,  a  theory  which  adapts  itself  with  surprising  facility  to  all 
the  observations  to  which  these  colors  can  be  submitted.     The 
number  and  importance  of  those  applications  account  sufficiently 
for  the  care  which  Newton  bestowed  on  his  experiments  on  co- 
lored rings.f 

796.  Among  the  experiments  of  Newton  on  colored  rings, 
none  are  more  interesting  than  those  which  he  instituted  on  soap 
bubbles.     It  is  well  known,  that  when  these  bubbles  are  inflated 
to  a  certain  degree  of  thinness,  very  gaudy  colors  make  their 
appearance,  and  hence  these  are  selected  as  favorite  objects  of 

*  This  phraseology  has  an  air  much  more  hypothetical  than  the  reality,  the  thing 
signified  being  little  more  than  the  simple  enunciation  of  a  fact  ascertained  by  ex. 
pertinent.  Probably  the  singularity  of  the  phrase  has  contributed  to  bring  the  doc- 
trine into  discredit,  or  even  into  ridicule,  with  those  who  have  never  looked  any  fur- 
ther into  it  than  to  read  the  title.  The  most  profound  opticians  of  modern  times, 
have  regarded  these  investigations  o'  Newton,  as  among  the  most  ingenious  and  sa- 
gacious of  all  his  labors. 

t  Biot. 


:  PTICS.  537 

amusement  for  children.  But  it  was  reserved  for  no  less  a  mind 
than  that  of  Newton,  to  make  these  exhibitions  the  means  of 
penetrating  the  secrets  of  nature. 

In  preparing  the  bubbles  for  experiment,  he  took  various  in- 
genious precautions  to  form  them  in  the  most  perfect  manner, 
and  preserved  them  for  deliberate  examination,  by  covering 
them  with  a  glass  receiver  which  protected  them  from  the  agi- 
tation of  the  air,  and  means  were  devised  for  preventing  any 
extraneous  light  from  mixing  with  that  of  the  bubble.  Things 
being  thus  arranged,  and  the  eye  placed  in  a  favorable  position, 
a  number  of  concentric  horizontal  rings  are  seen,  exhibiting  vivid 
colors  disposed  with  perfect  regularity.  They  correspond  in  ap- 
pearance to  those  exhibited  by  the  plate  of  air  between  the 
lenses,  (Art.  790,)  but  are  more  elegant  and  perfect  in  every  re^- 
spect.  Similar  exhibitions  of  color  are  presented  in  glass  bub- 
bles blown  exceedingly  thin ;  and  also  in  thin  laminae  of  the 
mineral  called  mica.  Analogous  variations  of  color  are  seen 
even  in  the  tarnish  of  certain  metals,  particularly  in  plates  of 
copper  and  steel  when  they  have  been  heated  in  the  open  air, 
and  they  appear  in  the  plumage  of  birds. 

797.  The  following  propositions,  several  of  which  have  al- 
ready been  incidentally  mentioned,  will  present  a  summary  of 
the  Newtonian  doctrine  of  colors. 

1.  The  colors  of  natural  bodies  are  not  qualities  inherent  in 
the  bodies  themselves,  by  which  they  immediately  affect  our 
sight,  but  are  a  mere  consequence  of  that  peculiar  disposition  of  the 
particles  of  each  body,  by  which  it  is  enabled  more  copiously  to  rejlect 
the  rays  of  one  particular  color,  and  to  transmit,  or  stifle,  or,  more 
properly,  to  absorb,  the  others. 

2.  The  colors  of  natural  bodies  are  the  colors  of  thin  plates, 
produced  by  the  same  cause  as  that  which  produces  them  in  thin 
laminae  of  air,  glass,  &c.,  viz.  the  interval  between  the  anterior 
and  posterior  surfaces  of  the  atoms.     The  thickness  of  the  atoms 
of  a  medium,  and  of  the  interstices  between  them,  determines 
the  color  they  reflect  or  transmit  at  a  particular  incidence,  be- 
cause it  must  depend  on  the  thickness  of  any  lamina,  whether 
the  light  when  it  has  reached  its  posterior  surface  is  in  the  state 
favorable  for  transmission  or  for  reflexion. 

3.  Opacity  in  natural  bodies  arises  from  the  multitude  of  reflex- 
ions caused  in  their  internal  parts.     By  this  means,  the  rays  are 
conceived  to  be  entangled,  as  it  were,  running  their  rounds  from 
atom  to  atom,  without  a  possibility  of  reaching  the  surface  and 
escaping. 

It  would  be  inconsistent  with  the  nature  of  an  elementary 
work  like  the  present,  to  enter  into  all  the  details  of  this  remark- 
able hypothesis  :  for  such  disquisitions  we  must  refer  the  student 
to  Brewster's  Optics,  to  Biot's  Traite  de  Physique,  to  Herschel's 

68 


NATURAL   PHILOSOPHY. 

Treatise  on  Light,  Professor  Bartlett's  Optics,  and  to  various  oth- 
er works  of  great  ability  which  have  been  written  on  these  sub- 
jects within  the  present  century. 

INFLEXION    OE   DIFFRACTION    OF   LIGHT. 

798.  INFLEXION  or  DIFFRACTION  is  a  term  used  to  denote  certain 
phenomena  which  light  exhibits,  when,  under  certain  circumstances, 
it  forms  parallel  bands  or  fringes. 

For  the  purpose  of  experiments  on  this  subject,  a  beam  of 
light  is  admitted  into  a  dark  room,  through  a  very  small  aper- 
ture, as  a  pin-hole  made  in  sheet  lead  ;  or,  what  is  better,  a  con- 
vex lens  is  placed  in  the  window  shutter,  which  brings  the  rays 
to  a  focus,  and  affords  a  divergent  pencil  of  light.  If  we  intro- 
duce into  this  pencil  any  opake  body,  as  a  knife-blade,  for  exam- 
ple, and  observe  the  shadow  which  it  casts  on  a  white  screen, 
we  shall  observe  on  both  sides  of  the  shadow  fringes  of  colored 
light,  the  different  colors  succeeding  each  other  in  the  following 
order :  first  fringe,  violet,  indigo,  pale  blue,  gr&n,  yellow,  red ; 
second,  blue,  yellow,  red  ;  third,  pale  blue,  pale  yellow,  red.  The 
brightness  of  these  fringes  diminishes  as  they  recede  from  the 
shadow,  and  the  shadow  itself  is  not  quite  dark,  but  is  formed 
also  of  luminous  and  dark  fringes,  all  parallel  to  the  edges  of  the 
knife.  The  fringes  in  question  are  absolutely  independent  of  the 
nature  of  the  body  whose  shadow  they  surround,  and  the  form 
of  its  edge ;  neither  the  density  or  rarity  of  the  one,  nor  the 
sharpness  or  curvature  of  the  other,  having  the  least  influence 
on  their  breadth,  their  colors,  or  their  distance  from  the  shadow. 
Thus  it  is  indifferent  whether  they  are  formed  by  the  edge  or 
back  of  a  razor,  by  a  mass  of  platina,  or  by  a  bubble  in  a  plate 
of  glass,  (which,  though  transparent,  yet  throws  a  shadow  by 
dispersing  away  the  light  incident  upon  it ;)  circumstances  which 
make  it  clear  that  their  origin  has  no  connection  with  the  ordi- 
nary refractive  powers  of  bodies,  or  with  any  elective  attractions 
or  repulsions  exerted  by  them  on  light ;  for  such  forces  cannot 
be  conceived  as  independent  of  the  density  of  the  body  exerting 
them,  however  minute  we  might  regard  the  sphere  of  their  ac- 
tion.* 

799.  If  the  light  of  the  solar  beam  be  first  separated  into  the 
prismatic  colors,  and  these  severally  be  submitted  to  experiment, 
the  fringes  will  in  each  case  be  of  the  same  color  as  the  colored 
pencil ;  but  they  will  be  broadest  in  red  light,  smallest  in  violet, 
and  of  intermediate  sizes  in  the  intermediate  colors.    If  we  place 
the  screen  at  different  distances  from  the  interposed  body  which 
gives  the  shadow,  it  will  be  found  that  the  fringes  grow  less  and 

*  Herschel.— firewater,  Life  of  Newton,  p.  103. 


ernes.  539 

less  as  we  approach  the  edge  of  the  body  from  which  they  take 
their  rise.  On  measuring  the  distances  of  the  fringes  from  the 
shadow,  while  they  are  thus  changing  their  dimensions,  and  con- 
necting by  a  line  the  several  points  representing  those  distances, 
it  is  found  that  this  line  is  not  a  straight  line,  but  a  hyperbola, 
whose  vertex  is  at  the  edge  of  the  body ;  so  that  the  same  fringe 
is  not  formed  of  the  same  light  at  all  distances  from  the  body, 
but  resembles  a  caustic  curve,  (Art.  742,)  formed  by  the  inter- 
section of  different  rays.  When  we  consider  that  the  fringes  are 
largest  in  red,  and  smallest  in  violet  light,  it  is  easy  to  understand 
the  cause  of  their  colors  in  white  light ;  for  the  colors  seen  in 
this  case  arise  from  the  superposition  of  fringes  of  all  the  seven 
colors  ;  that  is,  if  the  eye  could  receive  all  the  differently  colored 
fringes  at  once,  these  colors  would  form  by  their  mixture  the 
actual  colors  in  the  fringes  seen  by  white  light.  Hence  we  see 
why  the  color  of  the  first  fringe  is  violet  near  the  shadow,  and 
red  at  a  greater  distance  ;  and  why  the  blending  of  the  colors 
beyond  the  third  fringe,  forms  white  light,  instead  of  exhibiting 
themselves  in  separate  tints. 

Upon  measuring  the  proportional  breadths  of  the  fringes  with 
great  care,  Newton  found  that  they  were  as  the  numbers  1,  V|, 
v/j,  V},  and  their  intervals  in  the  same  proportion.* 

800.  In  the  foregoing  experiments,  the  colored  fringes  are  sup- 
posed to  be  formed  on  the  edge  of  the  shadow  of  an  opake  body, 
placed  in  a  beam  or  pencil  of  light.  The  same  phenomena  are 
exhibited  in  a  more  striking  and  beautiful  manner,  when  we  view 
with  a  magnifying  glass  a  pencil  of  light  as  it  passes  through 
an  exceedingly  small  aperture.  Suppose,  for  instance,  we  place 
a  sheet  of  lead,  having  a  small  pin-hole  pierced  through  it, 
in  the  pencil  of  rays  diverging  from  the  focus  of  a  lens.  The 
image  of  the  hole  will  be  seen  through  the  magnifier  as  a 
brilliant  spot,  encircled  by  rings  of  colors  of  great  vividness, 
which  contract  and  dilate,  and  undergo  a  singular  and  beautiful 
alternation  of  tints,  as  the  distance  of  the  hole  from  the  luminous 
point  on  the  one  hand,  or  from  the  magnifier  on  the  other,  is 
changed.  When  the  latter  distance  is  considerable,  the  central 
spot  is  white,  and  the  rings  follow  nearly  the  orders  of  colors  of  their 
plates.  When  the  magnifier  is  brought  very  near  to  the  pin-hole 
the  central  white  spot  contracts  into  a  point  and  vanishes,  and 
the  rings  gradually  close  in  upon  it  in  succession,  so  that  the 
center  assumes,  successively,  the  most  surprisingly  vivid  and  in- 
tense hues,  and  the  rings  surrounding  it  undergo  great  and  ab- 
rupt changes  in  their  tints,  f 

Newton  attempted  to  account  for  the  inflexion  of  light  by  a 
supposed  repulsion  exerted  by  the  edge  of  the  interposed  body 

*  Brewster's  Optics,  pp.  95 — 97.  t  Herschel  on  Light,  Sec.  730. 


640  NATURAL   PHILOSOPHY. 

or  by  the  edges  of  the  circular  aperture  on  the  rays  of  light  that 
are  nearest  to  it,  while  they  exert  a  less  repulsion  on  such  as  are 
a  little  more  remote.  By  this  means,  the  relative  direction  of 
the  rays  would  be  so  altered  that  they  would  cross  one  another, 
and  their  light  interfere  or  become  blended ;  and  by  following 
out  the  consequences  of  this  interference,  they  are  found  to 
correspond  to  some  of  the  effects  actually  observed  to  take 
place.* 

801.  A  more  satisfactory  explanation  of  the  inflexion  of  light 
and  the  formation  of  colored  fringes,  is  afforded  by  that  theory 
which  considers  light  as  produced,  not  by  the  emission  of  luminous 
particles  from  the  radiant  body,  but  by  the  undulations  of  a  peculiar 
fluid.  Phenomena  of  the  foregoing  description  are  accounted  for 
on  the  doctrine  of  interferences.  It  appears  by  experiment,  that  a 
body  already  illuminated  may  become  less  bright  by  the  addition  of 
more  light.  Let  the  solar  light,  reflected  horizontally,  be  admit- 
ted into  a  dark  chamber  by  two  small  holes,  which  are  near  each 
other,  but  separated  by  such  an  interval  that  the  conical  pencils 
do  not  intermix  until  they  have  proceeded  a  certain  distance. 
A  little  beyond  the  point  where  they  intermix,  let  them  be  re- 
ceived on  a  screen ;  then  at  some  points  of  the  illumined  part 
there  will  be  a  partial  or  comparative  darkness.  If  now,  one  of 
the  openings  is  closed,  so  that  the  light  is  not  intermixed  on  the 
illumined  part,  but  the  screen  receives  light  only  from  one  hole, 
the  partial  darkness  will  vanish,  and  parts  of  the  remaining  cir- 
cle will  have  become  brighter  by  this  loss  of  light.  Here  it  is  evi- 
dent that  the  addition  of  fresh  light  produces  darkness,  and  that 
an  obscure  surface  becomes  brighter  by  the  removal  of  some  of 
the  light  which  shines  upon  it. 

When  the  preceding  experiment  is  made  with  great  care,  and 
light  of  one  color,  as  red  or  blue,  is  admitted  through  two  fine 
slits  or  holes  near  each  other,  the  pencil  produces  bands,  which 
are  alternately  bright  and  dark,  exactly  analogous  to  the  bands 
produced  in  the  experiments  on  diffraction.  But  when  either  of 
the  apertures  is  closed,  the  bands  disappear,  and  the  space  in 
which  they  were  is  occupied  by  light  that  is  nearly  uniform. 
Thus  the  stoppage  of  the  light  from  one  aperture  removes  the 
partial  obscurity  which  existed  between  the  bright  spaces.  This 
darkness,  therefore,  results  from  the  concourse  of  the  two 
lights  meeting  obliquely  from  the  two  apertures.  Hence  it  ap- 
pears that  two  homogeneous  rays  of  light,  emanating  from  the 
same  source,  may,  after  passing  over  a  certain  distance,  come  to 
a  point  under  such  circumstances  that  the  brightness  will  be  al- 
most annihilated.  This  effect  can  be  referred  to  nothing  else  but 

*  Fresnel,  however,  has  shown  that  this  phenomenon  is  independent  of  the  edges 


OPTICS.  541 

the  mutual  action  of  the  rays  of  light,  and  to  this  mutual  action 
the  term  interference  has  been  applied. 

Something  quite  analogous  to  this  occurs  in  the  phenomena  of 
sound.  The  transmission  of  sounds  being  by  undulations,  we 
may  conceive  two  undulations  to  exist  exactly  similar,  and  to 
produce  simultaneous  impulses  in  the  same  direction,  so  that  the 
effect  on  the  ear  will  be  double  what  either  would  have  produced 
separately.  Here  one  sound-wave  is  augmented  by  the  addition 
of  another.  But  we  may  also  readily  conceive  cases  in  which 
one  sound-wave  may  interfere  with  another,  so  that  the  combined 
effect  is  less  than  either  would  have  produced  alone.  Effects  of 
this  kind,  well  known  in  music,  are  called  beats  ;  and  this  and 
numerous  other  analogies  between  the  phenomena  of  light  and 
those  of  sound,  which  have  been  traced  with  the  most  refined 
ingenuity  by  several  eminent  opticians,  have  produced  a  strong 
impression  in  favor  of  the  undulatory  theory  of  light.* 


CHAPTER  VII. 

OF  DOUBLE  REFRACTION  AND  POLARIZATION. 

802.  DOUBLE  REFRACTION  is  a  modification  which  light  undergoes 
in  passing  through  certain  media,  by  which  a  single  pencil  of  light 
is  divided  into  two  pencils,  affording  two  separate  images  of  the  ob- 
jects. 

This  phenomenon  was  first  observed  in  a  crystal  Fig.  285. 

•of  carbonate  of  lime,  denominated  Iceland  spar. 
This  substance  may  be  seen  in  every  cabinet  of 
minerals,  presenting  the  figure  of  a  rhomb.  It  is 
a  solid,  bounded  by  six  rhomboidal  faces.  It  is 
colorless  and  highly  transparent,  and  distinguish- 
ed for  its  beauty  in  mineralogical  collections  ;  but  its  most  re- 
markable property  is  that  of  rendering  letters,  or  any  other  small 
objects  placed  behind  it,  double. 

Though  double  refraction  is  exhibited  by  the  Iceland  crystal 
in  a  manner  peculiarly  striking,  yet  this  phenomenon  is  by  no 
means  confined  to  that  substance.  It  takes  place  in  numerous 
transparent  crystals.  It  also  occurs  in  a  variety  of  other  bodies, 
which  are  more  or  less  transparent,  where  there  is  any  disposi- 
tion toward  a  regular  arrangement  of  the  particles,  such  as 
hair,  quills,  and  the  like,  and  in  all  bodies  when  in  a  state  of 
unequal  dilatation  or  compression. 

*  See  the  explanation  in  full  in  Pouillet's  El.  de  Phys.,  ii,  106. 


542  NATURAL   PHILOSOPHY. 

Some  classes  of  bodies  possess  the  property  of  double  refrac- 
tion naturally  and  permanently ;  others  may  be  made  to  acquire 
it  by  artificial  means,  but  retain  it  only  transiently. 

803.  The  explanation  of  this  singular  effect  has  exercised  the 
sagacity  of  the  profoundest  philosophers,  at  the  head  of  whom 
are  Newton  and  Huygens.     Inquiries  respecting  it  have  of  late 
years  been  associated  with  those  respecting  the  polarization  of 
light,  both  of  which  subjects  have  been  studied  with  the  greatest 
attention  and  zeal  by  some  of  the  first  philosophers  of  the  present 
century ;  and  their  investigations  have  opened  a  new  field  of 
philosophical  curiosity,  no  less  ample  than  fertile.     In  a  work  so 
limited  as  the  present,  it  will  be  impossible  to  give  any  thing 
more  than  a  very  slight  sketch  of  these  subjects,  to  serve  merely 
for  the  purposes  of  an  introduction  to  studies  which,  in  order  to 
be  fully  understood,  require  to  be  prosecuted  for  a  length  of  time 
proportioned  to  their  extent  and  intricacy.     Unimportant  as  these 
researches  might  appear,  on  a  superficial  view,  they  are,  never- 
theless, in  common  with  several  other  refined  inquiries  in  optics, 
highly  conducive  to  that  full  knowledge  of  the  properties  of  light, 
which  enables  the  artist  to  give  perfection  to  such  noble  instru- 
ments as  the  microscope  and  the  telescope. 

804.  If  a  rhomb  of  Iceland  spar, 
represented  in   Fig.    286,   be  placed 
above  a  black  line  drawn   on  white 
paper,  and  viewed  with  the  eye  at  R, 
the  line  will  appear  double,  as  mn, 
MN  ;  or  if  we  cause  a  pencil  of  light 
Rr,   to   fall  upon  the  surface  of  the 
rhomb,  it  will  be  separated  into  two 
pencils,  rO,  rE,  each  of  which  will 
emerge  from  the  rhomb  at  o'  and  e  in  °'-- 
directions  Oo',  Ee',  parallel  to  Rr.    The   e' 

pencil  Rr  has  therefore  suffered  double  refraction  in  passing 
through  the  rhomb.  This  might  be  suspected  to  arise  from  a 
difference  in  density  of  the  different  parts,  since  the  parts  which 
were  more  dense  would  refract  the  light  more  than  those  which 
were  less  dense,  and  thus  tend  to  produce  separate  images ;  but 
as  the  same  effects  will  take  place  by  making  the  pencil  Rr  fall 
at  the  same  incidence,  and  in  the  same  direction,  relative  to  the 
summit  A  upon  any  point  of  any  of  the  faces,  it  is  manifest  that 
the  double  refraction  cannot  arise  from  any  difference  of  density 
in  different  parts  of  the  rhomb. 

805.  To  illustrate  the  manner  in  which  these  effects  take  place, 
let  AMXN,  (Fig.  287,)  be  a  section  of  the  Iceland  crystal,  (Fig. 


M  E'    0'     E  O 


B   O   X 


cerics.  543 

285,)  made  by  a  plane  passing  through  the  diagonal  AX.  Let 
PV  be  a  perpendicular  to  the  upper  surface  of  the  crystal  at  any 
point  F.  Then  rays  of  light  DF,  CF,  will  each  have  two  re- 
fracted rays,  FO,  FO',  and  FE,  FE'.  On  measuring  the  several 
angles,  it  will  be  found  that  the  two  rays  FO,  FO',  follow  the  or- 
dinary law  of  refraction,. making  the  sines  of  the  angles  of  re- 
fraction as  the  sines  of  those 
of  incidence.  Each  of  these, 
therefore,  is  called  the  ordi- 
nary ray.  But  the  other 
rays  FE,  FE',  will  not  con- 
form to  that  law,  but  will 
make  angles  with  the  per- 
pendicular, that  are  some- 
times greater  and  some- 
times less  than  would  be 
required  by  it,  and  are  thus 
separated  from  the  ordina- 
ry rays.  Hence,  each  of  these  is  called  the  extraordinary  ray. 
In  the  case  of  the  Iceland  crystal,  while  at  a  perpendicular  in- 
cidence the  ordinary  ray  undergoes  no  refraction,  the  extraordi- 
nary ray  is  turned  from  the  perpendicular  6°  12';  and  at  angles 
of  10°,  20°,  30°,  &c.,  while  the  ordinary  ray  suffers  a  regular 
refraction  according  to  the  law  of  the  sines,  the  extraordinary 
ray  suffers  a  refraction  constantly  greater  than  that. 

806.  The  line  which  connects  the  two  obtuse  angles  of  the 
rhomboid,  as  AX,  (Fig.  287,)  is  called  the  optic  axis,  since  when 
a  ray  enters  a  crystal  in  the  direction  of  this  line,  either  coinci- 
dent with  it  or  parallel  to  it,  such  a  ray  suffers  no  double  refrac- 
tion.    Thus,  the  ray  IF,  which  meets  the  crystal  in  such  a  direc- 
tion as  to  be  refracted  into  the  line  FK,  parallel  to  AX,  is  not 
separated  into  two  rays,  but  merely  undergoes  ordinary  refrac- 
tion.    When  the  extraordinary  ray  is  between  the  ordinary  ray 
and  the  perpendicular,  as  Fe,  the  crystal  is  said  to  have  a  positive 
axis  :  when  the  extraordinary  ray  is  further  from  the  perpendic- 
ular than  the  ordinary  ray,  it  is  said  to  have  a  negative  axis. 
Thus,  quartz  has  a  positive,  and  Iceland  spar  a  negative  axis. 

Any  plane,  like  that  of  Fig.  287,  passing  through  the  optic 
axis,  and  every  plane  parallel  to  this,  is  called  the  principal  sec- 
tion. Whenever  the  principal  section  contains  the  ordinary  ray, 
it  also  contains  the  extraordinary  ray,  which  is  not  the  case  with 
planes  inclined  to  that. 

807.  An  axis  of  double  refraction,  however,  is  not,  like  the 
axis  of  the  earth,  a  fixed  line,  within  the  rhomb  or  crystal.     It  is 
only  a  fixed  direction  :  for  if  we  divide,  as  we  may  do,  the  rhomb 
ABC  (Fig.  285)  into  two  or  more  rhombs,  each  of  these  separate 


544  NATURAL   PHILOSOPHY. 

rhombs  will  have  its  axis  of  double  refraction  ;  but  when  these 
rhombs  are  again  put  together,  their  axes  will  all  be  parallel  to 
AX.  Every  line,  therefore,  within  the  rhomb  parallel  to  AX,  is 
an  axis  of  double  refraction  or  optic  axis  ;  but  as  these  lines  have 
all  one  and  the  same  direction  in  space,  the  crystal  is  still  said  to 
have  only  one  optic  axis. 

Herschel,  in  his  Treatise  on  Light,  illustrates  this  subject  by 
the  following  simile.  Suppose  a  mass  of  brick  work,  or  mason- 
ry, of  great  magnitude,  built  of  bricks  all  laid  parallel  to  each 
other.  Its  exterior  form  may  be  what  we  please ;  a  cube,  a 
pyramid,  or  any  other  figure.  We  may  cut  it  (when  hardened 
into  a  compact  mass)  into  any  shape,  a  sphere,  a  cone,  a  cylin- 
der, &c.,  but  the  edges  of  the  bricks  within  it,  still  lie  parallel  to 
each  other  ;  and  their  directions,  as  well  as  those  of  the  diago- 
nals of  their  surfaces,  or  of  their  solid  figures,  may  all  be  re- 
garded as  so  many  axes,  i.  e.  lines  having  (so  long  as  the  mass 
remains  at  rest)  a  determinate  position,  or  rather  direction  in 
space,  no  way  related  to  the  exterior  surfaces,  or  linear  bounda- 
ries of  the  mass,  which  may  cut  across  the  edges  of  the  bricks 
in  any  angles  we  please. 

808.  A  great  number  of  crystals  have  two  axes  of  double  re- 
fraction, or  two  directions  inclined  to  each  other,  along  which 
the  double  refraction  is  nothing.  In  crystals  with  one  axis,  the 
axis  has  the  same  position,  whatever  be  the  color  of  the  pencil 
of  light  which  is  used  ;  but  in  crystals  with  two  axes,  the  axes 
change  their  position  according  to  the  color  of  the  light  employed, 
so  that  the  inclination  of  the  two  axes  varies  with  differently 
colored  rays. 

Until  recently  it  was  supposed  that  the  number  of  optic  axes 
never  exceeds  two  ;*  but  Dr.  Brewster  has  lately  discovered  an 
example  of  a  mineral  (analcime)  which  has  an  indefinite  number 
of  axes  of  double  refraction,  in  the  direction  of  which,  light  suf- 
fers no  separation,  although  when  passing  through  the  body  in 
any  other  direction,  it  undergoes  double  refraction. 

A  cylinder  of  glass,  first  heated  red  hot,  and  then  rolled  on  a 
plate  of  metal  until  it  is  cold,  acquires  a  permanent  doubly  re- 
fracting structure.  If,  instead  of  heating  the  glass  cylinder,  we 
had  placed  it  in  a  vessel,  and  surrounded  it  with  boiling  oil  or 
boiling  water,  it  would  have  acquired  the  same  doubly  refracting 
structure,  when  the  heat  had  reached  the  axis  ;  but  this  structure 
is  only  transient,  as  it  disappears  when  the  cylinder  is  uniformly 
heated.  Analogous  structures  may  be  produced  by  pressure, 
and  by  the  induration  of  soft  solids,  such  as  animal  jellies,  isin- 
glass, &c. 

*  Herschel  on  Light,  Sec.  781. 


OPTICS.  545 

809.  If  the  cylinder  in  the  preceding  article  is  not  a  regular 
one,  but  has  its  section  perpendicular  to  the  axis  everywhere  an 
ellipse  instead  of  a  circle,  it  will  have  two  axes  of  double  refrac- 
tion. In  like  manner,  if  we  use  rectangular  plates  of  glass,  in- 
stead of  cylinders,  in  the  preceding  experiment,  we  shall  have 
plates  with  two  planes  of  double  refraction  ;  a  positive  structure 
being  on  one  side  of  each  plane  and  a  negative  on  the  other.  If 
we  use  perfect  spheres,  there  will  be  axes  of  double  refraction 
along  every  diameter,  and  consequently  an  infinite  number  of 
them.  The  crystalline  lenses  of  the  eyes  of  almost  all  animals, 
whether  their  figures  be  those  of  lenses,  spheres,  or  spheroids, 
have  on«  or  more  axes  of  double  refraction. 


POLARIZATION   OF   LIGHT. 

810.  POLARIZATION  OF  LIGHT  is  a  change  which  light  undergoes 
after  certain  refractions  or  reflexions,  by  which  a  ray  acquires  PO- 
LARITY, or  different  properties  on  different  sides. 

This  quality  of  light,  which  is  one  of  the  most  remarkable  of 
all  its  properties,  was  discovered  by  Huygens,  during  his  investi- 
gations into  the  cause  of  double  refraction  as  exhibited  in  the 
Iceland  crystal ;  but  the  attention  of  opticians  was  more  particu- 
larly directed  toward  it  by  the  discoveries  of  Malus,  in  1810.* 
The  knowledge  of  this  singular  property^of  light  has  afforded 
an  explanation  of  many  of  the  most  intricate  phenomena  in  op- 
tics. 

811.  With  respect  to  the  light  of  the  sun,  whether  it  be  direct 
or  reflected,  whether  it  be  white  light  or  one  of  the  prismatic 
colors,  no  such  difference  of  properties  exists  in  the  different 
sides  of  a  ray  ;  and  the  same  is  true  of  the  light  of  a  candle  or 
any  self-luminous  body.      But  if  instead  of  employing  a  ray 
emitted  directly  from  the  sun  or  from  any  self-luminous  source, 
we  subject  to  examination  a  ray  that  has  undergone  double  re- 
fraction, or  a  certain  kind  of  reflexion  to  be  more  particularly 
described  hereafter,  or  that  has  been  in  any  one  of  a  great  vari- 
ety of  ways  subjected  to  the  action  of  material  bodies,  it  seeim 
to  have  acquired  sides  ;  a  right  and  a  left,  a  front  and  a  back  ? 
and  the  intensity,  or  brightness,  though  not  the  direction  of  the 
reflected  or  transmitted  portion,  depends  materially  on  the  posi- 
tion with  respect  to  these  sides,  in  which  the  plane  of  incidence 
lies,  though  every  thing  else  remains  precisely  the  same.f 

812.  We  may  understand  something  of  the  nature  of  the 

*  See  an  interesting  history  of  the  progress  of  these  discoveries,  in  the  Edinburgh 
Philosophical  Journal,  Vols.  I,  II,  &-c. 
t  Herschel 

69 


646  NATURAL    PHILOSOPHY. 

changes  produced  in  light  by  polarization  from  the  following  ex- 
periments. 

First,  we  place  on  a  horizontal  table  a  piece  of  black  paper, 
and  draw  on  it  two  fine  lines  at  right  angles  to  each  other,  hav- 
ing a  white  point  or  dot  at  their  intersection.  For  the  conve- 
nience of  reference,  we  will  suppose  that  the  direction  of  one  of 
these  lines  is  north  and  south,  and  that  of  the  other  east  and 
west.  Over  these  lines  we  place  a  crystal  of  Iceland  spar,  (Fig. 
285,)  having  its  principal  section  (which  it  must  be  recollected  is 
the  plane  passing  through  both  the  rays  into  which  the  incident 
light  is  divided)  in  the  meridian.  Agreeably  to  what  has  been 
said  in  Art.  805,  we  shall  see  two  images  of  the  white  spot 
formed  by  double  refraction.  The  light  thus  divided  will,  when 
transmitted  through  another  medium  producing  double  refrac- 
tion, become  polarized.  For,  if  we  place  a  second  crystal  of  the 
same  kind  directly  over  the  first,  having  their  principal  sections 
parallel,  or  coincident,  the  two  images  will  appear  as  before,  ex- 
cept farther  asunder.  But,  on  turning  the  upper  crystal  round, 
on  its  vertical  axis,  t\vo  new  images  begin  to  make  their  ap- 
pearance, being  very  faint  at  first,  but  growing  brighter,  while 
the  two  original  images  grow  weaker,  until  the  crystal  is  turned 
through  the  first  quadrant,  when  the  four  images  become  equally 
bright.  Continuing  to  turn  the  crystal  through  the  second  quad- 
rant, the  reverse  takes  place,  that  is,  the  new  images  grow  faint- 
er and  the  original  ones  brighter,  until  at  the  ekd  of  this  quad- 
rant, when  the  crystal  has  changed  sides,  only  the  two  original 
images  appear,  as  at  first.  Similar  changes  occur  in  turning  the 
crystal  through  the  third  and  fourth  quadrants.  Now  the  only 
change  of  circumstances  involved  in  the  foregoing  process,  is 
that  the  plane  of  incidence  is  successively  presented  to  the  dif- 
ferent sides  of  the  rays  of  light,  the  effect  being  greatest  when 
it  is  applied  to  the  sides  directly  opposite  to  each  other  ;  and  as 
a  particle  of  iron  filings,  however  small,  acquires,  when  mag- 
netized, different  and  peculiar  properties  in  two  opposite  points 
called  the  poles,  so  a  ray  of  light,  by  the  foregoing  process,  ac- 
quires properties  on  the  opposite  sides  somewhat  analogous  to 
polarity,  and  hence  is  said  to  be  polarized.* 

813.  Secondly,  for  an  example  of  polarization  by  reflexion,  we 
may  take  two  tubes,  a  larger  and  a  smaller,  the  latter  turning 
within  the  former,  like  two  tubes  of  a  hand  telescope.  The  com- 
pound tube  thus  formed  being  open  at  both  ends,  we  may  attach 
to  each  a  glass  reflecting  plate,  in  such  a  manner,  that  by  turn- 
ing round  the  smaller  tube  the  two  reflecting  plates  may  be 
placed  in  various  positions  with  respect  to  each  other.  Thus, 
(Fig.  288,)  let  MNP  be  the  tube,  A  and  C  plates  of  glass,  so  situ- 

*  Arago,  Fncyc.  Brit.,  Sup    Vol.  VI. 


OPTICS.  547 

ated  that  a  ray  of  iight  RA,  incident  at  A,  at  an  angle  of  56°, 
may  be  reflected  along  the  axis  of  the  tube  AC,  and  striking  on 
C  at  the  same  angle  of  incidence  of  56°,  be  reflected  to  the  eye 


at  E.  Then  in  the  position  shown  in  the  figure,  where  the  first 
reflexion  is  made  in  a  horizontal  plane  RAG,  and  the  second  in  a 
vertical  plane  ACE,  the  image  of  R  will  be  scarcely  visible  ;  but 
on  turning  round  the  tube  NP,  the  image  will  grow  brighter  and 
brighter,  until  the  plate  C  has  been  turned  round  ninety  degrees, 
when  it  becomes  the  brightest  possible.  On  being  turned  through 
the  second  quadrant,  the  image  grows  fainter  and  fainter,  until, 
at  180°  from  the  original  position  of  C,  the  image  almost  disap- 
pears. Similar  changes  occur  in  the  third  and  fourth  quadrants. 
Now,  at  the  commencement  of  the  experiment,  the  mirror  C  is 
applied  to  the  under  side  of  the  ray,  and  at  the  end  of  1 80°  it  is 
applied  to  the  upper  side,  at  both  which  points  there  is  scarcely 
any  reflexion  ;  but  when  the  same  plate  is  applied  laterally,  the 
reflexion  is  the  same  as  for  common  light.  Hence  this  light  had 
acquired  peculiar  properties  on  its  opposite  sides,  in  consequence 
of  its  previous  reflexion  from  the  first  plate  of  glass,  which,  as 
before,  from  its  analogy  to  magnetic  polarity,  is  denominated 
polarization.* 

814.  The  angle  of  56°  is  found  by  experiment  to  be  that  at 
which  this  effect  was  produced  by  glass,  and  hence  this  is  called 
the  polarizing  angle  for  glass.  Other  substances  have  different 
polarizing  angles.  Thus,  while  the  two  reflecting  plates  are  in 
the  position  shown  in  the  figure,  in  which  case  there  is  scarcely 
any  reflexion  from  the  second  plate,  and  consequently  no  image 
formed  at  the  eye,  yet  on  moistening  the  plate  C,  even  with  the 
breath,  the  image  instantly  appears,  since  the  polarizing  angle 
for  water  is  different  from  that  of  glass. 

Although  the  images  formed  by  polarized  light  are  variable  in 

*  Library  of  Useful  Knowledge. 


548  NATURAL   PHILOSOPHY. 

brightness,  yet  the  direction  of  such  rays  is  the  same  as  in  com- 
mon light,  and  consequently  no  change  is  produced  by  polariza- 
tion in  the  place  of  the  image.* 

815.  The  relation  of  color  to  polarized  light  is  highly  interest- 
ing, the  most  gorgeous  and  varied  hues  being  developed  in  ex- 
periments on  this  subject.  The  following  is  a  simple  example. 

Let  two  plates  of  glass  be  placed  as  in  Fig.  289,  so  that  the 
light,  which  may  be  that  of  the  sky,  will  be  polarized  by  reflex- 
ion from  the  plate  A,  in  consequence  of  which  the  eye  situated 
at  E,  will  not  see  the  image  of  the  sky  reflected  from  C,  as  it 
would  do  in  the  case  of  common  light,  but  in  the  place  of  it  will 
see  a  dark,  undefined  spot.  But  on  interposing  between  the  two 
plates  a  thin  film  of  selenite,f  or  mica,  the  eye  being  still  at  E, 

Fig.  289. 


we  shall  see  the  surface  of  the  interposed  plate  covered  with  the 
most  brilliant  colors.  If  its  thickness  is  perfectly  uniform 
throughout,  the  tint  will  be  uniform  ;  but  if  it  has  different 
thicknesses,  every  different  thickness  will  display  a  different 
color, — some  red,  some  green,  some  blue,  some  yellow,  and  all 
of  the  most  brilliant  description.  If  we  turn  the  plate  EDFG 
round,  keeping  it  perpendicular  to  the  polarized  beam,  the  colors 
will  become  less  or  more  bright,  without  changing  their  nature, 
and  two  lines  ED,  FG,  at  right  angles  to  each  other,  will  be 
found  such,  that  when  either  of  them  is  in  the  plane  of  reflexion 
RAC,  no  colors  whatever  are  perceived,  but  the  undefined  dark 
spot  before  mentioned  returns  again.  On  continuing  the  rotation 
of  the  plate,  the  colors  reappear,  and  reach  their  greatest  bright 
ness  when  either  of  the  lines  ab  or  cd,  which  are  inclined  4-5°  to 
the  lines  ED,  FG,  is  in  the  plane  of  reflexion  ACE. 

Let  the  plate  EDFG  be  now  fixed  in  the  position  where  it 

»  See  a  good  explanation  of  these  phenomena  in  Herschel's  Treatise  on  Light, 
Sects.  816—819. 

t  A  variety  of  sulphate  of  lime  or  plaster  of  Paris 


OPTICS.  549 

gives  the  brightest  color,  which  suppose  to  be  red,  and  let  the 
plate  C  be  made  to  revolve.  The  brightness  of  the  red  color 
will  gradually  decline  until  the  plate  has  turned  round  45°,  when 
it  will  wholly  disappear,  and  the  dark  spot  be  seen  again.  Be- 
yond 45°  a  green  color  will  gradually  make  its  appearance,  and 
will  become  brighter  and  brighter  until  it  reaches  its  maximum 
at  90°.  Beyond  90°  the  green  fades,  and  disappears  at  135°, 
when  the  dark  spot  returns  ;  but  beyond  this  the  red  reappears, 
and  reaches  its  maximum  at  180°.  Hence,  when  the  plate 
EDFG  alone  revolves,  only  one  color  is  seen,  and  when  C  only 
revolves,  two  colors  are  seen  during  each  half  of  its  revolution. 
If  we  repeat  these  experiments  with  plates  of  different  thick- 
nesses in  different  parts,  giving  different  colors,  we  shall  find 
that  the  two  colors  are  always  complementary  of  each  other,  or 
together  make  white  light.*  The  experiment  may  be  varied  so 
as  to  evolve  the  red  and  green  colors  at  the  same  time,  making 
one  overlap  the  other :  the  parts  thus  united  form  a  perfect 
white.f 

Instead  of  the  plate  of  selenite  or  mica,  by  exposing  doubly 
refracting  crystals  of  different  substances  to  the  action  of  polar- 
ized light,  an  endless  variety  of  beautiful  colors,  arranged  in  the 
most  fanciful  forms,  may  be  seen,  specimens  of  which  are  usu- 
ally represented  by  figures,  in  works  that  treat  at  large  of  this 
subject.  J 


CHAPTER  VIII. 

OF  VISION. 

816.  As  a  preparation  for  studying  the  optical  structure  of  the 
eye  and  the  laws  of  vision,  it  will  be  useful  first  to  learn  in  what 
way  images  of  external  objects  are  formed  in  a  dark  room,  by 
light  admitted  through  a  hole  in  the  window  shutter.. 

817.  A  beam  of  light  from  the  sun,  entering  into  a  dark  room 
through  a  small  orifice,  and  striking  upon  an  opposite  wall  or  screen, 
forms  a  circular  image  on  the  wall,  whatever  be  the  shape  of  the 

orifice. 

We  will  suppose  the  orifice  to  be  comparatively  large,  as  an 
inch  in  diameter,  and  of  a  triangular  or  of  an  irregular  shape  ; 
the  image  formed  on  the  wall  will  still  be  circular.  For,  suppose 
the  orifice  to  be  reduced  to  a  very  small  circular  hole,  as  a  pin- 

*  Brewster's  Optics,  159.  t  Library  of  Useful  Knowledge, 

t  See,  especially,  Pouillet,  de  Phys.,  t.  II. 


550  NATURAL   PHILOSOPHY. 

hole,  (which  may  easily  be  done  by  placing  over  the  orifice  a 
metallic  plate,  as  a  sheet  of  lead,  pierced  by  a  pin  ;)  then  the 
rays  of  the  sun,  passing  through  this  small  opening,  would  of 
course  be  circular.  But  the  large  irregular  orifice  may  be  con- 
sidered as  made  up  of  such  smaller  apertures,  or  the  metallic  plate 
may  be  conceived  to  be  pierced  with  an  indefinite  number  of  pin 
holes,  and  the  entire  image  formed  upon  the  wall  may  be  con- 
ceived to  be  made  up  of  an  assemblage  of  all  these  images  of 
the  sun  blended  with  each  other,  and  therefore,  as  bounded  by 
innumerable  curve  lines,  composed  of  the  individual  circles. 

If  the  screen  be  brought  near  to  the  ori-  Fig-  290' 

fice,  however,  the  image  will  be  of  the  same 
figure  as  the  orifice  ;  for  the  rays,  after  they 
have  passed  the  orifice,  must  have  diverged 
considerably  before  the  sections  that  form 
the  image  shall  afford  circles  so  large,  that 
their  blended  circumferences  shall  compose 
a  circular  figure.  (See  Fig.  290.) 

If  the  plane  which  receives  the  image  be 
not  parallel  to  the  orifice,  then  the  image 
will  be  elliptical,  being  the  section  of  a  cone  oblique  to  its  axis. 

Circular  images  of  the  sun  are  sometimes  projected  on  the 
ground,  through  the  small  openings  among  the  leaves  of  the  trees. 
During  an  eclipse  of  the  sun,  these  images  copy  the  figure  of 
the  eclipse. 

If  there  are  various  orifices  near  to  each  other,  three,  for  exam- 
ple, through  which  a  beam  of  the  sun  shines  into  a  dark  room, 
we  shall  observe  at  first,  at  a  certain  distance,  three  distinct  lu- 
minous circles.  At  a  greater  distance,  these  three  circles  begin 
to  be  blended,  and  finally,  on  enlarging  sufficiently,  they  unite  to 
form  a  single  circle.* 

818.  If,  instead  of  a  beam  of  solar  light,  we  admit  into  a  dark 
room,  through  an  opening  in  the  shutter,  the  light  reflected  from  va- 
rious objects  without,  an  inverted  picture  of  these  objects  will  be  form- 
ed on  the  opposite  wall. 

A  room  fitted  for  exhibiting  such  a  picture  is  called  a  Camera 
Obscura. 

From  what  has  been  before  explained,  it  will  be  readily  un- 
derstood, that  from  every  point  in  the  object,  innumerable  rays 
of  light  proceed  and  fall  upon  the  window  shutter.  Of  these, 
however,  none  can  enter  the  aperture,  except  such  as  are  very 
near  to  each  other,  all  others  diverging  too  far  to  enter  a  small 
opening.  It  is  essential  to  the  distinctness  of  the  picture,  that 
rays  which  proceed  from  every  point  in  the  object,  should  be  col- 
lected into  corresponding  points  in  the  image,  and  should  exist 

*  Barlow,  in  Encyc.  Metropol.     Art.  Optics 


OPTICS.  551 

there  free  from  any  mixture  of  rays  from  any  other  point ;  and  it 
is  essential  to  the  brightness  of  the  picture,  that  as  many  rays  as 
possible  should  be  conveyed  from  each  point  in  the  object  to  its 
corresponding  point  in  the  image.  To  render  the  picture  dis- 
tinct, therefore,  the  opening  in  the  window  shutter  must  be 
small,  else  the  pencils  of  rays  from  different  points  will  overlap 
each  other,  and  confuse  the  picture  ;  but  as  the  orifice  is  dimin- 
ished, the  brightness  of  the  picture  is  impaired,  since,  in  this 
case,  a  smaller  number  of  rays  are  conveyed  from  the  object  to 
the  image. 

These  modifications  of  the  picture  according  to  the  size  of  the 
aperture,  may  be  easily  exhibited  by  beginning  with  a  circular 
aperture,  two  or  three  inches  in  diameter,  and  reducing!  its  size 
gradually,  by  covering  it  with  a  piece  of  board,  or  a  metallic 
plate,  perforated  with  holes  of  different  sizes.* 

819.  If,  instead  of  passing  through  the  naked  orifice,  the  rays  be 
received  on  a  convex  lens,  two  or  three  inches  in  diameter,  Jixed  in 
the  window  shutter,  a  very  bright  and  distinct  picture  of  the  exter- 
nal landscape  will  be  formed  on  a  screen,  placed  at  the  focal  dis- 
tance of  the  lens. 

The  image  is  brighter  and  more  distinct  than  when  formed 
without  the  aid  of  the  lens ;  first,  because  the  diameter  of  the 
lens  may  be  so  great  as  to  receive  and  transmit  a  much  larger 
portion  of  the  rays  which  proceed  from  each  point  of  the  object, 
than  would  be  compatible  with  distinctness,  if  so  large  a  naked 
aperture  were  employed ;  secondly,  because  the  rays  of  each  pen- 
cil are  brought  more  accurately  to  a  separate  focus  ;  and  thirdly, 
because  the  picture  being  formed  nearer  to  the  window  shutter, 
it  is  smaller,  and  of  course  the  light,  being  spread  over  less  space, 
is  more  intense. 

A  convex  lens  fixed  in  a  ball  is  used  for  this  purpose,  which  is 
so  attached  to  the  opening  in  the  shutter,  as  to  be  capable  of  being 
turned  toward  different  parts  of  the  landscape,  like  the  eye-ball 
in  its  socket.  Such  a  lens,  with  its  accompanying  parts,  is  called 
a  Scioptic  ball. 

In  a  bright  sunny  day,  when  the  sun  is  on  the  side  of  the 
house  opposite  to  the  shutter,  and  of  course  illuminating  the  sides 
of  objects  which  face  the  window,  we  may  form,  either  with  or 
without  the  aid  of  the  scioptic  ball,  a  very  striking  and  beautiful 

*  A  small  room,  ten  feet  square  for  example,  having  a  window  opening  toward  an 
unobstructed  landscape,  may  easily  be  converted  into  a  camera  obscura.  The  perfo- 
ration in  the  shutter  must  be  made  equidistant  from  the  sides  of  the  room  ;  and  from 
the  aperture  as  a  center,  with  a  radius  equal  to  the  distance  of  the  opposite  wall,  de- 
scribe an  arc  of  a  circle,  upon  which,  as  a  base,  a  new  concave  wall  is  to  be  con- 
structed, finished  with  stucco.  The  other  walls  and  ceiling  are  to  be  colored  a  dead 
black,  while  the  concave  wall,  for  receiving  the  image,  is  made  as  white  as  possible. 
On  admitting  the  light  through  an  aperture  half  an  inch  in  diameter,  a  beautiful  and 
distinct  picture  will  be  formed  on  the  opposite  wall. 


552  NATURAL   PHILOSOPHY. 

picture  of  external  objects,  exhibiting  each  in  its  relative  situa- 
tion, of  a  size  and  brightness  corresponding  to  its  distance,  with 
all  the  colors  and  the  most  delicate  motions  of  the  landscape. 
The  name  camera  obscura,  which  appropriately  belongs  to  such 
a  chamber,  is  also  extended  to  certain  boxes,  in  which  similar 
pictures  are  formed,  with  peculiar  devices  for  rendering  the  image 
erect  instead  of  inverted.  The  Daguerreotype  is  an  instrument 
by  means  of  which  the  image  formed  on  the  principle  of  the  ca- 
mera obscura,  is  fixed  so  as  to  be  permanent. 

The  eye  is  a  camera  obscura,  and  the  analogy  existing  between 
its  principal  parts  and  the  contrivances  employed  to  form  a  picture 
of  external  objects,  as  in  the  preceding  experiments,  will  appear 
very  striking  on  comparison. 

820.  The  EYE*  is  an  assemblage  of  lenses  which  concentrate 
the  rays  emanating  from  each  point  of  external  objects  on  a  deli- 
cate tissue  of  nerves,  called  the  retina,  there  forming  an  image 
or  exact  representation  of  every  object,  which  is  the  thing  im- 
mediately perceived,  or  felt  by  the  retina.     Figure  291,  is  a  sec- 
tion of  the  human  eye  through  its  axis,  in  a  horizontal  plane. 
Its  figure  is,  generally  speaking,  spherical,  but  in  front  considera- 
bly more  prominent  than  the  corresponding  portion  of  a  sphere. 
The  eye  consists  of  three  principal  chambers,  filled  with  media 
of  perfect  transparency,  whose  refractive  powers  differ  consider- 
ably among  themselves,  but  none  of  them  is  greatly  different  from 
pure  water.     The  first  of  these  media,  Fig.  291. 

A,  occupying  the  anterior  chamber,  is 

called  the  aqueous  humor,  and  consists, 

in  fact,  chiefly  of  pure  water,  holding 

in  solution  a  little  common  salt  and 

gelatine,  with  a  trace  of  albumen.     Its 

refractive  index,  (Art.  747,)  is  almost 

precisely  that  of  water,  viz.  1.337,  that 

of   water  being    1.336.     The   cell   in 

which  the  aqueous  humor  is  contained, 

is  bounded,  on  its  anterior  side,  by  a  strong,  horny,  and  delicate 

ly  transparent  coat,  aa,  and  is  called  the  cornea,  the  figure  of 

which  is  an  ellipsoid,  produced  by  the  revolution  of  an  ellipse 

about  its  major  axis. 

821.  We  have  seen,  (Art.  761,)  that  convex  lenses  of  a  spheri- 
cal curvature  do  not  bring  rays  of  light  accurately  to  a  focus, 
but  spread  the  light  over  a  space  of  greater  or  less  extent,  which 
is  called  the  spherical  aberration  of  the  lens.     It  has  also  ap- 
peared, (Art.  763,)  that  if  the  lens  be  made  of  the  figure  pro- 

*  The  subjoined  description  of  the  eye  is  taken  chiefly  from  Herschel's  Treatise  on 
Light. 


OPTICS.  553 

duced  by  the  revolution  of  an  ellipse  on  its  major  axis, — an  el- 
lipse whose  major  axis  is  to  the  distance  between  the  foci,  as  the 
sine  of  incidence  is  to  the  sine  of  refraction  ;  then  the  rays  will 
be  brought  accurately  to  a  focus,  and  no  spherical  aberration 
will  take  place.  We  have  before  us,  in  the  aqueous  humor,  an 
example  of  this  construction.  Its  figure  is  such  an  ellipsoid,  the 
ratio  of  whose  major  axis  to  the  distance  between  the  foci,  is  al- 
most precisely  the  same  with  that  which  exists  between  the  sines 
of  incidence  and  refraction  ;  the  former  ratio  being  expressed 
by  1.3,  and  the  latter  by  1.337;  hence  parallel  rays  incident  on 
the  cornea  in  the  direction  of  its  axis,  are  made  to  converge  to 
a  focus  situated  behind  the  cornea,  with  almost  mathematical 
precision,  the  aberration,  which  would  have  occurred  had  the 
external  surface  been  a  spherical  figure,  being  almost  complete- 
ly destroyed. 

822.  At  the  posterior  surface  of  the  chamber  A  of  the  aqueous 
humor,  is  the  iris  cc,  which  is  a  kind  of  circular  opake  screen,  or 
diaphragm,  consisting  of  muscular  fibres,  by  whose  contraction  or 
expansion,  an  aperture  in  its  center,  called  the  pupil,  is  diminished 
or  dilated,  according  to  the  intensity  of  the  light.  In  very  strong 
lights,  the  opening  of  the  pupil  is  greatly  contracted,  so  as  not  to 
exceed  twelve  hundredths  of  an  inch  in  the  human  eye,  while 
in  feebler  illuminations  it  dilates  to  an  opening  not  exceeding 
twenty-five  hundredths,  or  double  its  former  diameter.  The  use 
of  this  is  evidently  to  moderate  and  equalize  the  illumination  of 
the  image  on  the  retina,  which  might  otherwise  injure  its  sensi- 
bility. In  animals,  as  the  cat,  which  see  well  in  the  dark,  the 
pupil  is  almost  totally  closed  in  the  daytime,  and  reduced  to  a 
very  narrow  line  ;  but  in  the  human  eye,  the  form  of  the  aperture 
is  always  circular.  The  contraction  of  the  pupil  is  involuntary, 
and  takes  place  by  the  effect  of  the  stimulus  of  the  light  itself ; 
a  beautiful  piece  of  self-adjusting  mechanism,  the  play  of  which 
may  be  easily  seen  by  bringing  a  candle  near  to  the  eye,  while 
directed  to  its  own  image  in  a  looking  glass.  Immediately  be- 
hind the  opening  of  the  iris,  lies  the  crystalline  lens  B,  enclosed 
in  its  capsule,  which  forms  the  posterior  boundary  of  the  cham- 
ber A.  The  figure  of  the  crystalline  lens  is  a  solid  of  revolu- 
tion, having  its  anterior  surface  much  less  curved  than  the  pos- 
terior. Both  figures  are  ellipsoids  of  revolution  about  their 
lesser  axes  ;  but  the  axes  of  the  two  figures  are  neither  exact- 
ly coincident  in  direction  with  each  other,  nor  with  that  of  the 
cornea.  This  deviation  would  be  fatal  to  distinct  vision,  were 
the  crystalline  lens  very  much  denser  than  the  others,  or  were 
the  whole  refraction  performed  by  it.  This,  however,  is  not  the 
case :  for  the  mean  refractive  index  of  the  lens  is  only  1.384, 
while  that  of  the  aqueous  humor,  as  we  have  seen,  is  1.337  ;  and 
that  of  the  vitreous  humor  C,  which  occupies  the  third  chamber, 
70 


554  NATURAL   PHILOSOPHY. 

is  1.339  ;  so  that  the  whole  amount  of  bending  which  the  rays 
undergo  at  the  surface  of  the  crystalline,  is  small  in  comparison 
with  the  inclination  of  the  surface  at  the  point  where  the  bend- 
ing takes  place  ;  and  since  near  the  vertex,  a  material  deviation 
in  the  direction  of  the  axis  can  produce  but  a  very  minute 
change  in  the  inclination  of  the  ray  to  the  surface,  the  cause  of 
error  is  so  weakened  in  its  effect,  as  probably  to  produce  no  ap- 
preciable aberration.  The  consistence  of  the  crystalline  is  that 
of  a  hard  jelly,  and  it  is  purer  and  more  transparent  than  the 
finest  rock  crystal. 

823.  In  the  crystalline,  a  very  curious  and  remarkable  contri- 
vance is  adopted  for  overcoming  or  preventing  the  spherical  ab- 
erration, which  (Art.  761)  belongs  to  lenses  of  this  form,  that 
refract  the  rays  more  toward  their  marginal  than  near  their  cen- 
tral parts,  and  hence  do  not  bring  all  the  rays  belonging  to  one 
pencil  to  the  same  focus.  Here  the  difficulty  is  obviated  by  giv- 
ing to  the  central  portions  of  the  crystalline  a  proportionately 
greater  density,  thus  increasing  its  refractive  power  so  as  exact- 
ly to  correspond  to  that  of  the  other  portions  of  the  lens. 

The  posterior  chamber  C  of  the  eye  is  filled  with  the  vitreous 
humor,  a  fluid  differing  neither  in  specific  gravity  nor  in  chemi- 
cal composition,  in  any  sensible  respect,  from  the  aqueous  ;  and 
as  we  have  already  seen,  having  a  refractive  index  but  little  su- 
perior to  it.  Its  name  is  derived  from  its  supposed  resemblance 
to  melted  glass ;  it  is  a  clear,  gelatinous  fluid,  very  much  re- 
sembling the  white  of  an  egg.  Rays  of  light  diverging  from  va- 
rious objects  without,  on  passing  through  the  aqueous  humor, 
(which  is  a  concavo-convex  lens,)  have  their  divergency  much 
diminished,  or  even,  in  most  cases,  are  rendered  converging,  and 
in  this  state  are  transmitted  through  the  crystalline,  which  has 
precisely  such  a  degree  of  refractive  power  as  enables  it  to  bring 
them  to  a  focus  at  the  distance  of  the  retina,  which,  as  a  screen, 
is  spread  out  to  receive  the  image.  The  retina,  as  its  name  im- 
ports, is  a  kind  of  white  net- work,  like  gauze,  formed  of  incon- 
ceivably delicate  nerves,  all  branching  from  one  great  nerve  O, 
called  the  optic  nerve,  which  enters  the  eye  obliquely  at  the  in- 
ner side  of  the  orbit,  next  the  nose.  The  retina  lines  the  whole 
of  the  cavity  C  up  to  ii,  where  the  capsule  of  the  crystalline 
commences.  Its  nerves  are  in  contact  with,  or  immersed,  in  the 
pigmentum  nigrum,  a  dark  velvety  matter,  which  covers  the  cho- 
roid  membrane,  mm,  and  whose  office  is  to  absorb  and  stifle  all 
the  light  which  enters  the  eye  as  soon  as  it  has  done  its  office  of 
exciting  the  retina ;  thus  preventing  internal  reflexions,  and 
consequent  confusion  of  vision.  The  whole  of  these,  humors  and 
membranes  are  contained  in  a  thick  tough  coat,  called  the  sclero- 
tica,  to  which  is  joined  the  cornea,  which  forms  what  is  called  the 
white  of  the  eye. 


OPTICS.  555 

824.  Such,  in  general,  is  the  structure  by  which  parallel  rays, 
and  those  coming  from  very  distant  objects,  are  brought  to  a  fo- 
cus on  the  retina.     But  there  are  special  contrivances,  suited  to 
particular  purposes,  which  are  no  less  evincive  of  design  and 
skill  than  the  general  organization  of  the  eye.     Some  of  the 
most  remarkable  of  these  we  proceed  to  mention.     The  cornea, 
by  protruding,  collects  the  rays  of  light  that  come  to  the  eye 
laterally,  and  guides  them  into  the  eye,  thus  enlarging  the  range 
of  vision.     It  answers  to  an  appendage  of  the  microscope,  which 
will  hereafter  be  described  under  the  name  of  field  glass.     The 
motion  of  the  eye-ball,  by  means  of  which  the  pupil  may  be 
turned  in  different  directions,  conduces  to  the  same  purpose. 
Hence,  notwithstanding  the  minuteness  of  the  aperture  which 
admits  the  light,  (and  it  must  be  small,  otherwise  the  image  will 
not  be  distinct,)  the  eye  may  take  in  at  once,  without  moving  the 
head,  a  horizontal  range  of  110°  and  a  vertical  range  of  120°  ; 
namely,  50°  above,  and  70°  below  a  horizontal  line.* 

825.  As  the  radiant  approaches  the  lens,  the  image  recedes 
from  it  on  the  other  side  ;  and  in  our  experiments  on  the  forma- 
tion of  images,  we  are  obliged  either  to  change  the  place  of  the 
screen  every  time  the  distance  of  the  radiant  is  altered,  or  to 
substitute  a  new  lens,  which  will  either  throw  back  the  image 
as  much  as  the  increased  distance  of  the  radiant  brings  it  for- 
ward, or  which  brings  the  image  as  much  nearer  as  the  altered 
place  of  the  radiant  tends  to  carry  it  off.     How  then  is  the  dis- 
tinctness of  the  image  maintained  in  the  eye,  notwithstanding 
the  immense  variety  in  the  distances  of  objects  ?     We  can  con- 
ceive of  but  two  ways  in  which  this  can  be  accomplished  :  either 
by  lengthening  or  shortening  the  diameter  of  the  eye  in  the  di- 
rection of  its  axis,  so  as  to  alter  the  distance  of  the  retina  from 
the  cornea  and  crystalline,  or  by  altering  the  curvature  of  the 
refracting  lenses  themselves,  increasing  their  convexity  for  near 
objects,  and  lessening  it  for  objects  that  are  more  remote.     Per- 
haps both  causes  may  operate  ;  but  the  effect  is  believed  to  be 
produced  chiefly  by  the  latter  cause,  namely,  change  of  figure  in 
the  refracting  lenses.     On  this  subject,  Sir  J.  Herschel  remarks, 
that  it  is  the  boast  of  science  to  have  been  able  to  trace  so  far 
the  refined  contrivances  of  this  most  admirable  organ  ;  not  its 
shame  to  find  something  still  concealed  from  its  scrutiny ;  for  how- 
ever anatomists  may  differ  on  points  of  structure,  or  physiologists 
dispute  on  modes  of  action,  there  is  that  in  what  we  do  under- 
stand of  the  formation  of  the  eye  so  similar,  and  yet  so  infinitely 
superior,  to  a  product  of  human  ingenuity,— such  thought,  such 
care,  such  refinement,  such  advantage  taken  of  the  properties  of 
natural  agents  used  as  mere  instruments  for  accomplishing  a 

*  Brewster. 


556  NATURAL   PHILOSOPHY. 

given  end,  as  force  upon  us  a  conviction  of  deliberate  choice  and 
premeditated  design,  more  strongly,  perhaps,  than  any  single 
contrivance  to  be  found,  whether  in  art  or  nature,  and  render  its 
study  an  object  of  the  deepest  interest. 

826.  Writers  on  comparative  anatomy,  express  the  highest  ad- 
miration of  the  adaptation  of  the  eyes  of  different  animals  to  the 
media  in  which  they  respectively  live,  and  to  the  peculiar  wants 
or  habits  of  each.     Thus  the  .crystalline  lens  of  the  fish  is  formed 
with  peculiar  reference'  to  the  refracting  properties  of  water.     In 
the  human  eye,  this  lens  has  a  refractive  power  only  a  little  great- 
er than  that  of  water  ;  but  since  tne  light  passes  out  of  a  much 
rarer  medium,  (air,)  such  a  density  is  sufficient  to  bring  the  rays 
to  a  focus  ;  but  were  the  density  of  the  crystalline  lens  in  the 
eye  of  the  fish  no  greater  than  in  the  human  eye,  receiving  the 
light  from  a  medium  (water)  almost  as  dense  as  itself,  it  would 
be  unable  to  give  that  change  of  direction  to  the  rays  which 
would  be  essential  to  distinct  vision.     But  provision  is  made  for 
this  exigency  by  giving  to  the  crystalline  lens  a  much  greater 
density,  and  of  course  a  higher  refracting  power,  which  enables 
it  completely  to  fulfil  its  purpose. 

Animals  which  have  occasion  to  see  in  the  dark,  as  the  owl 
and  the  cat,  have  the  power  of  opening  or  closing  the  pupil  to  a 
much  greater  extent  than  man.  By  this  means,  they  are  enabled 
in  the  dark,  to  collect  a  far  greater  number  of  rays  of  light. 
But  as  such  an  expansion  of  the  pupil  would,  in  broad  daylight, 
endanger  the  safety  of  eyes  of  such  peculiar  delicacy,  the  iris 
closes  over  the  aperture  and  diminishes  it  with  every  increase  in 
the  intensity  of  the  light,  a  change  which  is  involuntary  on  the 
part  of  the  animal.  In  animals,  as  birds  which  pounce  upon 
their  prey,  the  pupil  of  the  eye  is  elongated  perpendicularly, 
while  in  those  that  ruminate,  as  the  ox,  it  is  elongated  horizon- 
tally ;  being,  in  each  case,  exactly  adapted  to  the  circumstances 
of  the  animal. 

827.  The  images  of  external  objects  are  of  course  formed  in- 
verted on  the  retina,  and  may  be  seen  there  by  dissecting  off  the 
posterior  coats  of  the  eye  of  a  newly  killed  animal,  as  an  ox,  and 
exposing  the  retina,  like  the  image  on  a  transparent  screen,  seen 
from  behind.     The  appearance  is  particularly  striking  and  beau- 
tiful when  the  eye  is  fixed  like  the  scioptic  ball,  in  the  window 
shutter  of  a  dark  room.     It  is  this  image,  and  this  only,  which  is 
felt  by  the  nerves  of  the  retina,  on  which  the  rays  of  light  act  as 
a  stimulus ;   and  the  impressions  therein  produced  are  thence 
conveyed  along  the  optic  nerve  to  the  sensorium,  in  a  manner 
which  we  must  rank  at  present  among  the  profound  mysteries  of 
physiology,  but  which  appear  to  differ  in  no  respect  from  that  in 
which  the   impressions  of  the  other  senses  are  transmitted 


OPTICS.  557 

Thus,  a  paralysis  of  the  optic  nerve  produces,  while  it  lasts,  to- 
tal blindness,  though  the  eye  remains  open,  and  the  lenses  retain 
their  transparency  ;  and  some  very  curious  cases  of  half  blind- 
ness have  been  successfully  referred  to  an  affection  of  one  of 
the  nerves  without  the  other.*  On  the  other  hand,  while  the 
nerves  retain  their  sensibility,  the  degree  of  perfection  of  vision 
is  exactly  commensurate  with  that  of  the  image  formed  on  the 
retina.  In  cases  of  cataract,  when  the  crystalline  lens  loses  its 
transparency,  the  light  is  prevented  from  reaching  the  retina,  or 
from  reaching  it  in  a  proper  state  of  regular  concentration  ;  be- 
ing stopped,  confused,  and  scattered,  by  the  opake  or  semi-opake 
portions  it  encounters  in  its  passage.  The  image,  in  consequence, 
is  either  altogether  obliterated,  or  rendered  dim  and  indistinct. 
If  the  opake  lens  be  extracted,  the  full  perception  of  light  re- 
turns ;  but  one  principal  instrument  for  producing  the  converg- 
ence of  the  rays  being  removed,  the  place  of  the  image,  in- 
stead of  being  on  the  retina,  is  considerably  behind  it,  and  the 
rays  being  received  on  it  in  a  state  of  convergence,  before  they 
are  brought  to  a  focus,  produce  no  regular  picture,  and  therefore 
no  distinct  vision.  But  if  we  give  to  the  rays,  before  they  enter 
the  eye,  only  a  moderate  degree  of  divergence,  by  the  applica- 
tion of  a  convex  lens,  so  as  to  render  the  lenses  of  the  eye  capa- 
ble of  finally  effecting  the  exact  convergence  of  the  rays  upon 
the  retina,  distinct  vision  is  the  immediate  result.  This  is  the 
reason  why  persons  who  have  undergone  the  operation  for  the 
cataract,  (which  consists  either  in  totally  removing,  or  in  putting 
out  of  the  way  an  opake  crystalline,)  wear  spectacles  unusually 
convex.  Such  glasses  perform  the  office  of  an  artificial  crystal- 
line. An  imperfection  of  vision  similar  to  that  produced  by  the 
removal  of  the  crystalline,  is  the  ordinary  effect  of  old  age,  and 
its  remedy  is  the  same.  In  aged  persons,  the  cornea  loses  some- 
thing of  its  convexity,  or  becomes  flatter.  The  refracting  power 
of  the  eye  is  by  this  means  diminished,  and  a  perfect  image  can 
no  longer  be  formed  on  the  retina,  the  point  to  which  the  con- 
verging rays  tend  being  beyond  the  retina.  The  deficient 
power  is  supplied  by  a  convex  lens,  in  a  pair  of  spectacles,  which 
are  so  selected  and  adapted  to  the  eye,  as  exactly  to  compensate 
for  the  want  of  refracting  power  in  the  eye  itself,  and  thus  the 
rays  are  brought  to  a  focus  at  the  retina,  where  alone  a  distinct 
image  can  be  formed. 

828.  Short-sighted  persons  have  their  eyes  too  convex,  forming 
the  image  too  soon,  or  before  the  rays  reach  the  retina.  Con- 
cave glasses  counteract  this  effect.  Rare  cases  have  occurred 
where  the  cornea  was  so  very  prominent  as  to  render  it  impos- 
sible to  apply,  conveniently,  a  lens  sufficiently  concave  to  coun- 

*  Wollaston,  Phil.  Trans.  1824. 


558  NATURAL   PHILOSOPHY. 

teract  its  action.  Such  cases  would  be  accompanied  with  im- 
mediate blindness,  but  for  that  happy  boldness,  justifiable  only 
by  the  certainty  of  our  knowledge  of  the  true  nature  and  laws 
of  vision,  which  in  such  a  case  has  suggested  the  opening  of  the 
eye  and  removal  of  the  crystalline  lens,  though  in  a  perfectly 
sound  state.*  Other  defects  of  eyesight,  whose  cause  has  been 
ascertained  to  depend  on  malconformation  of  the  cornea,  or  some 
other  part  of  the  eye,  have  sometimes  been  remedied  by  adapt- 
ing to  them  glasses  of  a  peculiar  construction,  possessing  optical 
properties  suited  to  the  particular  defects  they  were  required  to 
remedy. 

829.   The  impression  made  by  light  remains  on  the  eye  for  a  short 
time  after  the  light  itself  is  withdrawn. 

The  case  of  a  stick  ignited  Fig.  292. 

at  the  end  and  whirled  in  the 
air  has  already  been  noticed. 
(Art.  779.)  Upon  the  same 
principle,  the  spokes  of  a  wheel, 
and  other  parts  of  machinery  in 
rapid  motion,  exhibit  continuous 
surfaces,  although  made  up  of 
parts  which  are  separated  from 
each  other  by  large  intervals. 
Lightning,  also,  and  fiery  mete- 
ors, appear  to  describe  long  lines  of  light  merely  because  their 
passage  through  the  atmosphere  is  so  rapid,  that  the  eye  does 
not  lose  the  impression  of  the  first  portions  until  the  last  are 
added.  The  amusing  toy  called  the  Thaumatrope,-f  depends  on 
the  same  principle.  An  example  of  it  is  exhibited  in  the  pre- 
ceding cut,  (Fig.  292,)  which  represents  a  circular  card,  on  one 
side  of  which  is  inscribed  a  chariot,  and  on  the  other  the  chariot- 
eer. To  opposite  sides  of  the  circumference  of  the  card  are  at- 
tached strings,  by  means  of  which,  taken  between  the  thumb  and 
finger  of  each  hand,  a  rapid  revolution  is  given  to  the  card,  bring- 
ing the  figures  on  the  opposite  sides  in  quick  succession  before 
the  eye.  When  the  motion  is  so  swift  that  the  eye  retains  the 
impression  of  both,  the  two  appear  united,  or  the  charioteer  ap- 
pears in  his  proper  place,  driving  the  chariot.  The  Phantasma- 
scope  consists  of  disks  bearing  on  their  margin  a  variety  of  fig- 
ures, which  are  so  related  to  each  other,  that  each  succeeding 
figure  shall  afford  a  continuation  of  the  preceding,  and  the 
whole  taken  together,  when  put  in  rapid  revolution,  shall  ex- 
hibit a  single  figure  performing  some  singular  or  amusing  feat. 
Thus  the  figure  might  commence  with  a  player  holding  a  violin, 
and  a  bow  which  he  is  just  beginning  to  draw  ;  the  second  view 

*  Herschel  on  Light,  Sec.  350 — 358.  t  &avi*a,  a  wonder,  and  rpcvu,  to  turn. 


OPTICS.  559 

might  represent  the  bow  as  drawn  a  little  ;  the  third-  still  more  ; 
and  the  whole  views  would  then  exhibit  the  usual  motions  of 
the  bow.  In  a  similar  manner  are  performed  dances,  feats  of 
horsemanship,  and  the  like. 

830.  As  we  have  two  eyes,  and  a  separate  image  of  every  ex- 
ternal object  is  formed  in  each,  it  may  be  asked,  why  we  do  not 
see  double  ? 

When  we  look  at  an  object,  we  direct  towards  it  the  optic  ax- 
is* of  each  eye,  and  see  most  distinctly  the  point  where  this  axis 
produced  meets  the  body.  In  looking  at  the  same  point  with 
both  eyes,  we  incline  them  so  as  to  make  the  two  axes  meet  in 
that  point :  we  therefore  see  this  point  in  the  same  place  with 
both  eyes,  and  it  appears  as  one,  the  image  being  brighter  than 
when  seen  with  one  eye.  •  If,  by  any  means,  the  optic  axes  are 
prevented  from  meeting  in  the  same  point,  double  vision  is  the 
consequence.  Thus  we  make  surrounding  objects  appear  double 
by  pressing  the  ball  of  one  eye  sideways  with  the  finger.  Those 
who  have  one  eye  distorted  by  a  blow,  see  double,  though  they 
sometimes  learn  by  habit  to  correct  the  defect,  even  while  the 
distortion  remains.  The  sense  of  touch  is  subject  to  similar  dis- 
tortion :  if  we  lay  the  middle  finger  across  the  fore  finger,  and 
apply  the  ends  of  both  fingers  to  any  object,  as  a  small  ball,  or 
the  end  of  the  nose,  the  object  appears  double.  A  similar  sepa- 
ration of  the  optic  axes,  with  a  similar  result,  takes  place  when 
we  hold  a  small  object,  as  a  pin,  in  front  of  the  eyes,  and  then 
direct  them  to  some  distant  object:  the  pin  appears  double. 
The  same  effect  is  produced,  when  we  look  intently  at  an  ob- 
ject near  the  eye,  and  attempt  at  the  same  time  to  catch  a  view 
of  a  remote  object :  the  latter  appears  double. 

831.  The  reason  why  objects  appear  erect,  while  their  images 
on  the  retina  are  inverted,  has  given  rise  to  much  discussion.    It 
seems,  however,  a  point  not  difficult  to  comprehend,  that  objects 
and  the  parts  of  objects,  should  appear  in  the  direction  in  which 
the  rays  of  light  emitted  from  them  come  to  the  eye ;  and,  ac- 
cordingly, that  those  which  come  from  the  top  and  bottom  of  the 
object  should  be  referred  to  those  points  respectively,  just  as  one 
sound  would  be  known  to  proceed  from  the  top  and  another  from 
the  bottom  of  a  high  tower,  merely  by  the  different  sensations 
which  they  excited  in  the  ear,  although  the  chain  of  vibrations 
from  the  top  should  strike  the  bottom,  and  those  from  the  bottom 
the  top  of  the  ear.     Indeed,  this  very  circumstance  might  be  that 
which  determined  the  relative  positions  of  the  two  points :  and 


*  The  optic  axis  is  the  axis  of  the  crystalline  lens,  or  a  line  passing  through  the 
center  of  the  crystalline  perpendicular  to  both  its  surfaces. 


560  NATURAL   PHILOSOPHY. 

if  these  sounds  presented  to  the  mind  a  picture  of  the  tower, 
they  would  represent  it  in  its  natural  erect  position. 

Very  minute  objects,  which  cannot  be  seen  by  direct  vision, 
may  sometimes  be  rendered  visible  by  looking  a  little  way  from 
them,  so  that  their  light  strikes  the  eye  obliquely.  Thus,  as- 
tronomers, in  viewing  the  smallest  stars  or  satellites  with  the 
telescope,  have  sometimes  been  able  in  this  manner  to  catch  a 
glimpse  of  them,  when  they  could  not  otherwise  be  seen. 

832.  The  estimation  of  the  DISTANCES  and  MAGNITUDES  of  objects 
is  not  dependent  on  optical  principles  alone,  but  the  information  af- 
forded by  the  eye,  is  taken  in  connection  with  various  circumstances 
that  influence  the  mind  in  judging  of  these  particulars. 

In  the  first  place,  we  judge  of  the  distance  of  an  object  by  the 
inclination  of  the  optic  axes,  which  is  greater  for  nearer  objects 
and  less  for  objects  more  remote.  But  beyond  a  certain  dis- 
tance, this  method  is  very  indeterminate,  since  great  intervals 
among  remote  objects  would  scarcely  affect  the  inclination  of 
these  axes.  In  the  second  place,  we  judge  of  distance  by  the 
apparent  magnitude  of  known  objects  ;  as  when  a  ship  of  large 
size,  or  a  high  mountain,  appears  comparatively  small,  we  refer 
it  to  a  great  distance.  We  are  also  frequently  deceived  in  our 
estimate  of  distance  when  we  are  approaching  large  objects,  as 
a  great  city  or  a  lofty  mountain  :  we  fancy  they  are  nearer  than 
they  actually  are.  In  the  third  place,  we  estimate  the  distance 
of  objects  by  the  degree  of  distinctness  of  the  parts  or  brightness 
of  the  colors.  Thus,  a  smoky  mountain  is  referred  to  a  great 
distance  ;*  a  mountain  whose  sides  are  precipitous  and  bare,  (es- 
pecially where  the  rocks  have  a  new  and  fresh  appearance  in 
consequence  of  having  been  quarried  for  use,)  appears  nearer 
than  the  reality ;  vessels,  or  steamboats,  seen  through  a  mist  in 
the  night,  have  sometimes  run  foul  of  each  other,  being  supposed 
by  the  pilots  to  be  much  further  off,  in  consequence  of  the  indis- 
tinctness of  their  appearance.  In  the  fourth  place,  our  estimate 
of  distance  is  affected  by  the  number  of  intervening  objects. 
Hence,  distances  upon  uneven  ground  do  not  appear  so  great  as 
upon  a  plain ;  for  the  valleys,  rivers,  and  other  objects  that  lie 
low,  are  many  of  them  lost  to  the  sight.  On  this  principle,  the 
breadth  of  a  river  appears  less  when  viewed  from  one  side  than 
from  the  center ;  a  ship  appears  nearer  than  the  truth  to  one 
unaccustomed  to  judge  of  distances  on  the  water ;  and  the  hori- 
zontal distance  of  the  sky  appears  much  greater  than  the  vertical 
distance,  whence  the  aerial  vault  does  not  present  the  appear- 


*  This  appearance  exhibits  the  true  color  of  the  atmosphere,  becoming  visible  in 
consequence  of  the  extent  of  the  medium,  and  the  dark  ground  which  the  mountain 
affords  upon  which  to  view  it. 


OPTICS.  561 

ance  of  a.  hollow  hemisphere,  but  of  such  a,  hemisphere  much 
flattened  in  the  zenith,  and  spread  out  at  the  horizon. 

833.  A  similar  variety  of  circumstances  affects  our  estimate 
of  the  magnitudes  of  bodies  seen  at  different  distances.     First, 
the  visual  angle,  that  is,  the  angle  subtended  by  the  object  at  the 
eye,  determines  the  size  of  objects  that  are  near  ;  but  it  is  scarce- 
ly any  guide  to  the  dimensions  of  remote  objects,  since  all  such 
objects  subtend  angles  at  the  eye  comparatively  very   small. 
Thus,  on  this  principle,  a  fly,  within  a  few  inches  of  the  eye, 
would  appear  larger  than  a  ship  of  war  seen  at  some  distance 
on  the  water.     A  giant,  nine  feet  in  height,  thirty  feet  off,  would 
appear  no  larger  than  a  child  three  feet  high,  seen  at  the  dis- 
tance of  ten  feet.     But  as  this  result  is  not  conformable  to  expe- 
rience, it  is  evident  that  we  must  have  means  of  judging  of  the 
magnitude  of  objects,  beside  that  derived  from  the  visual  angle. 
If  the  giant  were  to  remove  from  the  distance  of  ten  feet  from 
the  eye  to  that  of  thirty  feet,  his  image  on  the  retina  would  bo 
only  one-third  as  long  as  before ;  but,  on  the  other  hand,  the  dis- 
tance is  trebled,  and  the  sort  of  combination  that  takes  place  in 
us  of  the  two  impressions,  the  one  of  magnitude,  the  other  of  dis- 
tance, is  like  the  constant  product  of  two  quantities,  of  which 
one  increases  in  the  same  ratio  as  the  other  diminishes  ;  whence 
the  giant  would  appear  constantly  of  the  same  height,  at  what- 
ever distance  from  us  he  was  seen.* 

834.  This  corrected  result,  however,  we  can  make  only  in  cases 
where  we  are  familiar  with  the  actual  size  of  the  body.     When 
not  thus  familiar,  we  rely  too  much  on  the  visual  angle,  and  are 
thus  often  greatly  deceived.     A  speck  on  the  window  being  at 
the  instant  supposed  to  be  an  object  on  a  distant  eminence,  is 
magnified,  in  our  estimation,  into  a  body  of  extraordinary  size, 
(as  a  line  half  an  inch  long  into  a  May-pole ;)  or  distant  objects 
supposed  to  be  very  near,  appear  of  an  exceedingly  diminutive 
size.     Secondly,  the  effect  of  contrast  is  visible  in  our  estimation 
of  the  magnitudes  of  bodies,  a  given  object  appearing  much  be- 
low its  ordinary  size,  when  seen  by  the  side  of  those  of  very  great 
magnitude.     Men  quarrying  stone  at  the  base  of  a  high  moun- 
tain, sometimes  appear,  at  a  little  distance,  like  pigmies,  partly 
from  the  effect  of  contrast,  but  more,  perhaps,  from  the  impres- 
sion which  the  mountain  gives  us  of  their  being  nearer  than  they 
actually  are.     Thirdly,  objects  seen  at  an  angle  considerably 
above  or  below  us,  as  a  man  on  the  top  of  a  spire,  or  a  river  in 
a  deep  valley  seen  from  the  top  of  a  mountain,  appear  greatly 
diminished.     In  these  cases,  since  there  are  no  intervening  ob- 
jects to  aid  us  in  estimating  the  distance,  we  estimate  it  too  low;. 

»  Hafiy. 

71 


562  NATURAL    PHILOSOPHY. 

and  hence  (Art.  832)  the  object  appears  less  than  the  reality. 
Moreover,  being  seen  obliquely,  its  apparent  dimensions  are  di- 
minished on  this  account,  the  apparent  diameter  being  deter- 
mined by  the  line  into  which  the  object  is  projected  perpendicu- 
lar to  the  axis  of  vision.  Hence,  children  judge  much  less  ac- 
curately both  of  distances  and  magnitudes  than  adults  ;  and 
blind  persons,  suddenly  restored  to  sight,  have  usually  displayed 
an  utter  inability  to  judge  of  these  particulars. 


CHAPTER  IX 

OF   MICROSCOPES 

835.  THE  Microscope  is  an  optical  instrument,  designed  to  aid 
the  eye  in  the  inspection  of  MINUTE  objects.* 

Telescopes,  on  the  other  hand,  assist  the  eye  in  the  examina- 
tion of  distant  bodies.  These  two  instruments  -  have  probably 
more  than  any  other  ex.tended  the  boundaries  of  human  thought, 
and  no  small  part  of  the  labor  which  has  been  bestowed  upon 
the  science  of  optics,  has  had  for  its  ultimate  object  their  im- 
provement and  perfection. 

With  the  hope  of  making  the  learner  well  acquainted  with 
the  principles  of  the  microscope,  we  shall  begin  with  those  vari- 
eties of  the  instrument  which  are  the  most  simple  in  their  con- 
struction, and  successively  advance  to  others  of  a  more  compli- 
cated structure. 

836.  The  simplest  microscope  is  a  double  convex  lens.     This, 
it  is  well  known,  when  applied  to  small  objects,  as  the  letters  of 
a  book,  renders  them  larger  and  more  distinct.     Let  us  see  in 
what  manner  these  effects  are  produced.     When  an  object  is 
brought  nearer  and  nearer  to  the  eye,  we  finally  reach  a  point 
within  which  vision  begins  to  grow  imperfect.     That  point  is 
called  the  limit  of  distinct  vision.     Its  distance  from  the  eye  va- 
ries a  little  in  different  persons,  but  averages  (for  minute  objects) 
about  jive  inches.     If  the  object  is  brought  nearer  than  this  dis- 
tance, the  rays  come  to  the  eye  too  diverging  for  the  lenses  of 
the  eye  to  bring  them  to  a  focus  soon  enough,  that  is,  so  as  to 
make  the  image  fall  exactly  on  the  retina.     Moreover,  the  rays 
which  proceed  from  the  extreme  parts  of  the  object  meet  the 
eye  too  obliquely  to  be  brought  to  the  same  focus  with  those 
rays  which  meet  it  more  directly,  and  hence  contribute  only  to 
confuse  the  picture.     We  may  verify  these  remarks  by  bringing 
gradually  toward  the  eye  a  printed   page   with  small  letters. 


OPTICS.  '  563 

When  the  letters  are  within  two  or  three  inches  of  the  eye,  they 
are  blended  together,  and  nothing  is  seen  distinctly.  If  we  now 
make  a  pin-hole  through  a  piece  of  paper,  (black  paper  is  pre- 
ferable,) and  look  at  the  same  letters  through  this,  we  find  them 
rendered  far  more  distinct  than  before  at  nearer  distances,  and 
larger  than  ordinary.  Their  greater  distinctness  is  owing  to 
the  exclusion  of  those  oblique  rays  which,  not  being  brought  by 
the  eye  to  an  accurate  focus  with  the  central  rays,  only  tend  to 
confuse  the  picture  formed  by  the  latter. 

As  only  the  central  rays  of  each  pencil  can  enter  so  small  an 
orifice,  the  picture  is  made  up,  as  it  were,  of  the  axes  of  all  the 
pencils.  The  increased  magnitude  of  the  letters  is  owing  to  their 
being  seen  nearer  than  ordinary,  and  thus  under  a  greater  angle, 
an  increase  of  the  visual  angle  having  much  influence  in  our 
estimate  of  the  magnitude  of  near  objects,  though  it  has  but  lit- 
tle influence  in  regard  to  remote  objects.  (Art.  833.) 

837.  A  convex  lens  acts  on  much  the  same  principle,  only  it 
is  still  more  effectual.  It  does  not  exclude  the  oblique  rays,  but 
it  diminishes  their  obliquity  so  much,  as  to  enable  the  eye  to 
bring  them  to  a  focus  upon  the  retina,  and  thus  to  make  them 
contribute  to  the  brightness  of  the  picture.  The  object  is  mag- 
nified as  before,  because  it  is  seen  nearer,  and  consequently  un- 
der a  larger  angle,  which  enables  minute  portions  to  be  distinct- 
ly recognized  by  the  eye,  which  were  before  invisible,  because 
they  did  not  occupy  a  sufficient  space  on  the  retina.  The  pow- 
er of  a  lens  to  accomplish  these  purposes,  will  obviously  depend 
on  its  refractive  power  ;  and  this  (supposing  the  material  of  which 
the  lens  is  made  to  remain  the  same)  will  depend  on  its  increas- 
ed sphericity,  and  diminished  focal  distance.  Lenses  of  the 
smallest  focal  distance,  therefore,  other  things  being  equal,  have 
the  greatest  magnifying  power,  and  spherules,  or  perfect  spheres, 
have  the  highest  magnifying  power  of  all.  When  the  radiant 
is  situated  in  the  focus  of  a  lens,  the  rays  go  out  parallel.  When 
thus  received  by  the  eye,  they  are  capable  of  being  brought  to  a 
focus  by  it,  and  of  forming  a  distinct  image.  Hence,  by  means 
of  a  lens,  an  object  may  be  seen  distinctly  when  it  is  exceeding- 
ly near  to  the  eye,  provided  it  be  situated  in  the  focus  of  the 
lens.  The  magnifying  power  of  a  lens,  therefore,  depends  on 
the  ratio  between  its  focal  distance  and  the  limit  of  distinct  vis- 
ion. The  latter  being  five  inches,  a  lens  whose  focal  distance  is 
one  inch,  by  bringing  the  object  five  times  nearer,  magnifies  its 
linear  dimensions  in  the  same  ratio,  and  its  superficial  dimen- 
sions in  the  ratio  of  the  square.  Thus  in  the  case  supposed,  an 
object  would  appear  five  times  as  long  and  broad,  and  have 
twenty-five  times  as  great  a  surface,  when  seen  through  the 
magnifier,  as  when  seen  by  the  naked  eye.  Lenses  have  been 
made  capable  of  affording  a  distinct  image  of  very  minute  ob- 


564  NATURAL   PHILOSOPHY. 

jects,  when  their  focal  distances  were  only  ¥'7  of  an  inch.  In 
this  case,  the  magnifying  power  would  be  ¥V  :  5»  which  is  as  1 
to  300,  or  as  1  to  90,000  in  surface. 

838.  When,  however,  an  object  is  so  near  to  the  eye,  a  very 
minute  space  covers  the  whole  field  of  vision,  and  it  is  only  the 
minutest  objects,  or  the  smallest  parts  of  a  body,  that  are  visible 
in  such  microscopes.     The  extent  of  parts  seen  by  a  microscope 
is  called  the  field  of  view.     A  microscope  of  small  focal  distance 
has  a  proportionally  small  field  of  view.     Moreover,  since,  when 
the  object  is  so  near  to  the  lens,  the  rays  of  light  strike  the  lens 
extremely  diverging,  only  the  central  rays  of  each  pencil  can  be 
brought  accurately  to  a  focus.    The  more  oblique  rays,  therefore, 
must  be  excluded  by  covering  up  all  but  the  central  portions  of 
the  lens,  by  which  means  the  brightness  of  the  image  is  dimin- 
ished.    The  part  of  a  lens  through  which  the  light  is  admitted, 
is  called  its  aperture.     The  aperture  of  a  lens  of  small  focal  dis- 
tance and  high  magnifying  powers,  must  of  necessity  be  small, 
and  one  of  the  principal  difficulties  in  the  use  of  such  micro- 
scopes, is  the  want  of  sufficient  light.     Hence  microscopes  of 
different   focal   distances   are  required  for  different   purposes. 
Where  we  wish  to  view  a  large  object  at  once,  we  must  use  a 
lens  which  has  a  large  field  of  view,  and  of  course  but  a  com- 
paratively small  magnifying  power.     Such  are  the  glasses  used 
by  watchmakers  and  other  artists.     Microscopes  which  magnify 
but  little,  yet  afford  a  large  field  of  view,  are  called  magnifiers, 
or  magnifying  glasses.     Such  are  the  large  lenses  employed  for 
viewing  pictures.     But  for  inspecting  the  minute  parts  of  a  small 
insect,  we  require  a  much  higher  power  ;  and,  the  object  being 
very  small,  a  large  field  of  view  is  not  necessary.     The  only  dif- 
ficulty to  be  obviated  is  the  want  of  light ;  and  this  evil  is  rem- 
edied, either  by  placing  the  object  in  the  sun,  or  by  condensing 
upon  it  a  still  stronger  light,  by  means  of  apparatus  specially 
adapted  to  that  purpose,  which  will  be  described  hereafter.* 

839.  Among  the  most  distinguished  achievements  of  philo- 
sophical artists,  in  our  own  times,  has  been  the  formation  of  mi- 
croscopes out  of  the  hardest  precious  gems,  especially  the  diamond 
and  the  sapphire.      The  diamond  seems  to  unite  in  itself  al- 
most every  desirable  quality  for  this  purpose.     It  will  be  recol- 
lected that  this  substance  is  distinguished  for  its  high  refractive 
powers,  its  index  of  refraction  being  2.439,  while  that  of  crown 
glass  is  only  1.530,  (Art.  748  ;)  hence  a  given  refracting,  and  of 
course  magnifying,  power  may  be  attained  with  a  lens  of  less 
curvature,  and  consequently  (Art.  761)  subject  to  less  spherical 

*  A  convenient  pocket  microscope  is  sometimes  sold  in  the  shops,  consisting  of  a 
slide  of  ivory  or  horn,  two  or  three  inches  in  length,  in  which  are  set  three  or  four 
lenses  of  different  powers,  adapted  to  various  purposes. 


OPTICS.  565 

aberration  than  glass  lenses  of  the  same  power.  Indeed,  it  is  es- 
timated that  the  indistinctness  arising  from  spherical  aberration 
in  a  diamond  lens,  is  only  -^\th  as  great  as  in  a  glass  lens  of 
equivalent  power.  The  sapphire  has  analogous  properties,  as 
also  the  garnet ;  and  pure  rock  crystal  (quartz)  is  much  es- 
teemed for  refracting  lenses  ;  but  some  of  the  pellucid  gems  are 
unsuitable  for  this  purpose  on  account  of  their  possessing  the 
property  of  double  refraction.  The  comparative  curvatures  and 
thicknesses  of  three  lenses  of  the  same  refracting  power,  made 
respectively  of  glass,  sapphire,  and  diamond,  are  exhibited  i»  the 
following  diagrams. 

Since,  moreover,  a  dia-  F'g-  293. 

mond  lens  admits  of  be- 
ing made  much  thinner 
than  a  glass  lens  of  the 

,       ,  ,,          Glass.  Sapphire.  Diamond. 

same  power,  the  loss  of 

light  by  absorption  is  far  less,  and  the  brightness  of  the  image  is 

proportionally  augmented. 

840.  Another  distinguishing  and  valuable  property  of  the  dia- 
mond is,  that  it  combines  with  a  high  refractive,  a  low  dispersive 
power.  By  dispersive  power  is  meant  the  power  of  separating  the 
different  colored  rays,  that  is,  of  decomposing  common  light  into  its 
prismatic  elements.  Diamond  lenses  are  naturally  nearly  achro- 
matic, or  afford  images  which  are  destitute  of  color.  But  while 
these  favorable  qualities  were  known  to  appertain  to  the  dia- 
mond, which,  taken  in  connection  with  its  great  transparency 
and  purity  of  structure,  were  observed  to  fit  it  admirably  for 
microscopes  of  great  magnifying  powers,  yet  the  extreme  hard- 
ness of  the  substance,  seemed  to  render  the  difficulty  of  grinding 
it  into  the  requisite  shape  almost  insuperable.  This  difficulty 
has,  however,  within  a  few  years,  been  completely  overcome  by 
Mr.  Pritchard,  an  eminent  English  artist,  who  has  constructed  a 
number  of  diamond  and  sapphire  microscopes,  whose  perform- 
ances have  equalled  the  most  sanguine  expectations. 

The  following  table  exhibits  the  different  magnifying  powers 
of  Pritchard's  sapphire  lenses,  corresponding  to  different  focal 
distances,  the  limit  of  distinct  vision  being  taken  at  T'r  of  an  inch. 

Magnifying  jower. 


Sort*  oT  an  inch. 

To' 

Linear. 

100 
150 

Superficial. 

10,000 
22  500 

200 
300 

40,000 
90  000 

400 
500 

160,000 
250  000 

>_--.. 

600 
700 

360,000 
490  000 

T1TT 

1000 

1,000,000 

566  NATURAL   PHILOSOPHY. 

841.  A  drop  of  transparent  liquor  may  be  easily  converted  in- 
to a  magnifier,  constituting  a  Fluid  Microscope.     The  simplest 
kind  of  fluid  microscope  is  formed  by  drilling  a  small  hole  in  a 
plate  of  brass  or  lead,  and  applying  to  it  a  drop  of  water  from 
the  point  of  a  pin.     If  the  plate  be  hollowed  out  on  both  sides 
around  the  aperture,  the  water  will  spontaneously  assume  the 
shape  of  a  convex   lens.     Water,  however,  possessing  only  a 
comparatively  low  refracting  power,  (Art.  748.)  is  less  adapted 
to  this  purpose  than  several  other  fluids,  particularly  certain 
transparent  balsams  and  aromatic  oils.     Sulphuric  acid  and  cas- 
tor oil  answer  well,  but  turpentine  varnish  and  Canada  balsam 
are  preferred,  especially  because  as  they  dry  they  become  indu- 
rated, and  form  permanent  microscopes.     Instead  of  the  aper- 
ture in  a  metallic  plate  above  described,  a  small  plate  of  glass 
may  be  employed,  in  which  case  it  is  only  necessary  to  drop  the 
varnish  or  balsam  on  the  surface  of  the  plate ;  and  it  will  as- 
sume the  figure  of  a  plano-convex  lens.     The  power  of  the  micro- 
scope may  be  varied  by  employing  a  larger  or  a  smaller  drop,  or 
by  suffering  it  to  spread  itself  on  the  upper  or  on  the  under  sur- 
face, since  the  curvature  of  the  drop,  and  of  course  its  focal 
distance,  is  modified  by  each  of  these  circumstances. 

842.  The  PERSPECTIVE  GLASS,  which  is  used  for  viewing  pic- 
tures, affords  another  example  of  the  application  of  the  simple 
microscope.     It  consists  of  a  large  double  convex  lens  fixed  in  a 
frame  in  a  vertical  position^from  the  top  of  which,  on  the  back 
side,  proceeds  a  plane  mirror,  which  is  fixed  at  an  angle  of  45° 
with  the  horizon,  and  of  course  it  makes  the  same  angle  with 
the  lens.     Pictures  to  be  viewed  are  placed  in  an  inverted  posi- 
tion, (that  is,  with  the  top  toward  the  spectator,)  on  a  table  at 
the  foot  of  the  instrument.     The  mirror  being  set  at  an  angle  of 
45°  with  the  horizon,  renders  horizontal  objects  erect.   (Art.  771.) 
Its  office,  therefore,  is  merely  to  give  a  proper  direction  to  the 
rays  of  light  from  the  picture  as  they  enter  the  lens,  causing 
them,  in  fact,  to  come  to  the  lens  in  the  same  manner  as  they  would 
do  were  the  mirror  removed  and  the  picture  set  up  in  a  vertical 
position,  parallel  to  the  lens,  at  a  distance  from  the  lens  equal  to 
the  length  of  any  ray,  measured  from  the  picture  to  the  mirror 
and  from  the  mirror  to  the  lens.     (Art.  729.)     Again,  in  order 
that  the  image  maybe  erect,  it  is ,  necessary  that  the  picture 
should  be  placed  with  its  top  toward  the  observer  ;  for  since  the 
image  of  every  point  in  the  picture  is  just  as  far  behind  the  mir- 
ror as  the  point  is  before  it,  those  parts  of  the  picture  which  are 
designed  to  occupy  the  highest  part  of  the  image  must  be  far- 
thest below  the  mirror.     This  will  be  understood  from  the  fol- 
lowing diagram. 

A  A,  a  convex  lens  fixed  vertically  in  a  frame. 

BB,  a  plane  mirror  making  with  the  horizon  an  angle  of  45° 


OPTICS. 


567 


C,  an  object  placed  horizontally  upon  the  table,  the  upper 
part  being  toward  the  observer. 

The  object  will  be  reflected  by  the  mirror  into  a  perpendicular 
position,  and  its  rays  will,  therefore,  fall  on  the  lens  in  the  same 
manner  as  they  would  were  no  mirror  Fig.  294. 

employed,  and  the  object  were  situated 
perpendicularly  behind  the  lens,  at  a 
distance  from  it  equal  to  the  sum  of 
the  distances  from  the  object  to  the 
mirror,  and  from  the  mirror  to  the  lens. 
Consequently,  if  the  distance  of  C 
from  the  lens  be  equal  to  the  focal  dis- 
tance of  the  lens,  the  rays  will  come 
to  the  eye  parallel,  and  a  distinct  and 
magnified  image  will  be  formed.  If 
the  distance  be  greater  than  the  focus, 
(as  it  may  be  rendered  by  depressing 
C  to  a  lower  level,)  then  the  rays  will 
come  to  the  eye  converging,  and  the 
image  will  be  more  magnified  but  less 
distinct.  If  the  distance  of  C  be  less 
than  the  focus,  the  image  will  be  less 
magnified,  but  it  will  be  distinct  with- 
in certain  limits.  The  reasons  of  these  several  modifications, 
will  be  evident  by  reflecting  on  principles  already  expounded. 
(Art.  754.) 

When  the  glass  is  of  good  quality,  and  the  picture  executed 
agreeably  to  the  rules  of  perspective,  the  various  parts  are  ex- 
hibited in  their  natural  positions,  and  at  their  relative  dis- 
tances, so  as  greatly  to  improve  the  view.  The  greater  distinct- 
ness of  the  parts  and  more  natural  distribution  of  light  and  shade 
than  what  attends  the  naked  view,  is  owing  not  only  to  the  in- 
creased magnitude  and  to  the  greater  quantity  of  the  light  emit- 
ted from  the  picture  which  is  collected  by  the  lens  and  conveyed 
to  the  eye,  but  also  to  the  separation  of  this  portion  of  light  from 
that  which  proceeds  from  various  other  objects.  The  lens  both 
conveys  more  of  the  light  of  the  picture  to  the  eye  than  would 
otherwise  reach  it,  and  conveys  it  unmingled  with  extraneous 
light.  The  importance  of  the  latter  circumstance  is  manifested 
even  by  looking  at  the  picture  through  an  open  tube,  or  through 
the  hand  so  curved  as  to  form  a  tube. 

843.  The  microscopes  hitherto  examined  are  such  as  are  de- 
signed to  be  interposed  between  the  eye  and  the  object  to  be 
viewed,  the  latter  being  placed  in  the  focus  of  parallel  rays  of 
the  lens,  or  a  little  nearer  to  the  lens  than  that  focus,  so  that  the 
rays  of  the  same  pencil  may  come  to  the  eye  either  parallel  or 
with  so  small  a  degree  of  divergency,  that  the  lenses  of  the  eye 


568  NATURAL   PHILOSOPHY. 

shall  be  competent  to  make  them  converge  and  form  an  image 
on  the  retina.  In  this  case,  as  the  rays  come  to  the  eye  in  the 
same  manner  as  rays  from  larger  objects,  at  a  greater  distance, 
seen  without  the  aid  of  a  lens,  the  position  of  the  object  is  not 
changed,  that  is,  it  is  seen  erect.  Single  microscopes,  however, 
are  also  employed  to  form  a  magnified  image  on  a  wall  or  screen, 
which  is  seen  by  the  eye  instead  of  the  object  itself.  Two  cele- 
brated instruments,  the  Magic  Lantern  and  the  Solar  Microscope, 
magnify  their  objects  in  this  manner,  in  the  construction  of  which, 
the  principles  under  review  are  happily  exemplified. 

844.  From  what  has  been  already  learned  respecting  lenses, 
the  following  points  will  be  readily  comprehended,  being,  for  the 
most  part,  a  recapitulation  of  principles  already  explained  and 
demonstrated. 

If,  in  a  dark  room,  we  place  before  a  convex  lens  any  lumin- 
ous object,  as  a  candle,  we  shall  observe  the  following  phenom- 
ena. (See  Art.  754.) 

1.  If  the  radiant  be  placed  nearer  to  the  lens  than  its  focus, 
since  the  rays  will  go  out  diverging,  no  image  will  be  formed  on 
the  other  side  of  the  lens. 

2.  Even  when  the  radiant  is  in  the  focus,  so  that  the  rays  go 
out  parallel,  they  never  meet  in  a  focus,  and  of  course  never 
form  an  image.* 

3.  But  when  the  radiant  is  further  from  the  lens  than  its  focus, 
the  rays  converge  on  the  other  side,  those  of  each  pencil  which 
proceed  from  the  same  point  in  the  object,  being  accurately 
united  in  one  point  of  the  image,  and  occupying  that  point  alone, 
without  the  interference  of  rays  from  any  other  point. 

4.  The  axes  of  the  rays  from  the  extreme  parts  of  the  object 
cross  each  other  in  the  center  of  the  lens.     Hence,  they  form  an 
image  inverted  with  respect  to  the  object ;  and  although  the  rays 
which  make  up  any  individual  pencil  are  made  to  converge  by 
the  lens,  yet  the  axes  (which  determine  the  magnitude  of  the 
picture)  diverge  from  each  other  after  crossing  at  the  center  of 
the  lens,  and  hence  the  image  is  greater  in  proportion  as  it  is 
formed  at  a  greater  distance  from  the  lens.     When  the  object  is 
only  a  little  further  off  from  the  lens  than  its  focus,  the  image  is 
thrown  to  a  great  distance,  and  is  proportionally  magnified.     As 
the  object  is  separated  further  from  the  lens,  (wh'ch  may  be  ef- 
fected either  by  withdrawing  the  object  from  the  lens  or  the  lens 
from  the  object,)  the  image  is  formed  at  a  less  distance,  and  is  of 
a  diameter  proportionally  less.     (See  Art.  760.)     Suppose  now 
that  we  employ  a  magnifier  of  so  small  focal  distance,  that  when 
the  object  is  placed  within  one  tenth  of  an  inch  of  the  lens,  the 
image  is  formed  on  the  other  side  upon  a  screen  or  wall,  at  the 

*  It  will  be  remarked,  that  when  the  single  microscope  is  used  as  an  eye-glass,  the 
eye  itself  brings  the  parallel  rays  to  a  focus  and  forms  the  image. 


OPTICS. 


569 


distance  of  twenty  feet ;  the  object  will  be  magnified  in  the  ra- 
tio of  TV  to  (20  x  12=)  240  ;  that  is,  the  image  will  be  2,400  times 
greater  than  the  object  in  diameter,  and  5,760,000  times  greater 
in  surface.  It  would  seem,  therefore,  as  if  nothing  more  were 
necessary,  in  order  to  form  magnified  images  of  objects,  than  a 
dark  room,  a  convex  lens,  and  a  screen  or  wall  for  the  reception 
of  the  picture.  It  must  be  remarked,  however,  that  when  the 
light  which  proceeds  from  the  object  is  diffused  over  so  great  a 
space,  its  intensit)'  must  be  greatly  diminished,  so  as  to  be  either 
incapable  of  affording  a  picture  which  shall  be  visible  at  all,  or 
at  least  sufficiently  bright  for  the  purposes  of  distinct  vision. 
This  difficulty  is  remedied  by  illuminating  the  object ;  and  it  is 
for  this  purpose  that  most  of  the  contrivances  employed  in  the 
magic  lantern  and  solar  microscope  are  designed. 

845.  The  MAGIC  LANTERN  consists  of  a  large  tin  canister,  either 
cylindrical  or  cubical  in  its  figure,  having  an  opening  near  the 
bottom  into  which  air  may  enter  freely  to  supply  the  lamp,  and 
a  chimney  proceeding  from  the  top,  and  bent  over  so  as  to  pre- 
vent the  light  of  the  lamp  from  shining  into  the  room.  The  lan- 
tern has  a  door  in  the  side,  which  shuts  close,  the  object  being 
throughout  to  prevent  any  light  from  escaping  into  the  room, 
except  what  attends  the  picture.  The  room  itself  is  made  as 
dark  as  possible ;  or,  what  is  better,  the  experiments  are  per- 
formed by  night.  In  front  of  the  lantern  is  fixed  a  large  tube, 
Fig.  295. 


at  the  open  end  of  which  is  placed  the  magnifying  lens.  In  the 
same  tube,  at  a  distance  from  the  lens  somewhat  greater  than 
the  focal  distance,  the  object  is  introduced,  which  is  usually 
some  figure  painted  on  glass  in  transparent  colors,  the  other 
parts  of  the  glass  being  blackened  so  that  no  light  can  pass 
through  except  that  which  falls  on  the  object  and  illuminates  it, 
by  which  means  we  shall  have  a  luminous  image  projected  on  a 
black  ground.  For  illuminating  the  object,  an  argand  lamp  is 
placed  near  the  center  of  the  lantern,  the  light  of  which  is  con- 
centrated upon  the  object  in  two  ways ;  first,  by  means  of  a 

72 


570  NATURAL   PHILOSOPHY. 

thick  lens,  usually  plano-convex,  so  situated  between  the  lamp 
and  the  object  that  the  rays  which  diverge  from  the  lamp  shall 
be  collected  and  condensed  upon  the  object ;  and,  secondly,  by 
means  of  a  concave  reflector,  situated  behind  the  lamp,  which 
serves  a  similar  purpose. 

A,  the  magnifying  lens. 

B,  the  object,  introduced  through  an  opening  in  the  tube. 

C,  the  condensing  lens. 

D,  the  lamp. 

E,  the  concave  mirror. 

F,  the  image  thrown  on  a  screen,  or  a  white  wall,  in  a  dark 
room. 

<z,  a  thumb-piece,  by  which  the  magnifier  may  be  made  to  ap- 
proach the  object  or  to  recede  from  it,  and  thus  the  image  be 
thrown  to  a  greater  or  less  distance,  according  to  the  magnitude 
required.  As  the  image  is  inverted  with  respect  to  the  object, 
it  is  only  necessary  to  introduce  the  object  itself  in  an  inverted 
position,  and  the  image  will  be  erect. 

The  objects  employed  in  the  Magic  Lantern  are  very  various, 
consisting  of  figures  of  men  and  animals ;  of  caricatures  ;  of 
representations  of  the  passions ;  of  landscapes  ;  and  of  astro- 
nomical diagrams.  When  the  last  are  employed,  this  apparatus 
becomes  subservient  to  a  useful  purpose  in  teaching  astronomy, 
and  is  frequently  so  employed  by  popular  lecturers  on  that  sub- 
ject. 

846.  The  SOLAR  MICROSCOPE  does  not  differ  in  principle  from 
the  Magic  Lantern,  only  the  object  is  illuminated  by  the  concen- 
trated light  of  the  sun  instead  of  that  of  a  lamp.*  And  sinc,e  a 
powerful  illumination  may  thus  be  effected  upon  minute  objects 
placed  before  a  magnifier  of  great  power,  the  solar  microscope 
is  usually  employed  to  form  very  enlarged  images  of  the  most 
minute  substances,  as  the  smallest  insects,  the  most  delicate  parts 
of  plants,  and  other  attenuated  objects  of  natural  history.  For 
magnifiers  several  of  different  focal  distances  are  employed,  va- 
rying from  an  inch  to  the  T\  or  ¥V  °f  an  inch,  it  being  understood 
that  those  of  the  shortest  focus  and  greatest  magnifying  powers 
can  be  used  only  for  the  minutest  objects,  since,  when  bodies  of 
a  larger  size  are  brought  so  near  a  small  lens,  their  light  strikes 
the  lens  too  obliquely  to  be  transmitted  through  it.  The  magni- 
fying lens  is  fixed  into  the  mouth  of  a  tube  and  the  object  placed 
near  its  focus,  much  in  the  same  manner  as  in  the  magic  lan- 
tern ;  but  instead  of  the  body  of  the  lantern,  (which  contains 
the  illuminating  apparatus,)  a  mirror,  about  three  or  four  inches 
wide,  and  from  twelve  to  eighteen  inches  long,  is  attached  to  the 

*  The  oxy-Jiydrogen  microscope  has  recently  been  substituted  for  the  solar,  the  in- 
tense flame  resulting  from  the  combustion  of  the  gaseous  elements  of  water,  being 
used  instead  of  the  sun's  light. 


OPTICS.  571 

other  end  of  the  tube.  This  mirror  is  thrust  through  an  opening 
in  the  window  shutter  of  a  dark  room,  and  the  mouth  of  the  tube 
to  which  it  is  fixed  is  secured  firmly  to  the  shutter,  so  that  the 
mirror  is  on  the  outside,  and  the  tube  with  its  lenses  is  on  the 
inside  of  the  shutter.  By  means  of  adjusting  screws,  the  mirror 
is  turned  in  such  a  way  as  to  direct  the  sun's  rays  into  the  tube, 
where  they  are  received  by  one  or  more  of  the  lenses  called  con- 
densers, which  collect  them  and  concentrate  them  upon  the  ob- 
ject, which  thus  becomes  highly  illuminated,  and  capable  of  af- 
fording an  image  sufficiently  bright  and  distinct,  though  magni- 
fied many  thousands  or  even  millions  of  times.  It  will  be  ob- 
served that  the  magnitude  of  the  image  depends  here,  as  in  other 
cases  of  the  simple  microscope,  upon  the  ratio  between  the  dis- 
tances of  the  object  and  the  image  from  the  center  of  the  mag- 
nifier. If,  for  example,  the  object  be  within  the  tenth  of  an  inch 
of  the  lens,  and  the  image  be  thirty  feet,  or  three  hundred  and 
sixty  inches  from  it,  then  the  image  will  be  360x10=3600  times 
as  large  as  the  object  in  diameter,  and  (3600)2=12,960,000  times 
in  surface.  With  a  given  lens,  the  size  of  the  image  depends 
wholly  on  the  distance  to  which  it  is  thrown  ;  that  is,  on  the  dis- 
tance of  the  wall  or  screen  where  it  is  formed. 

847.  When  the  solar  microscope  is  well  constructed,  it  affords 
the  most  wonderful  results,  and  greatly  enlarges  our  conceptions 
of  the  delicacy,  perfection,  and  subtilty  of  the  works  of  nature. 
In  inspecting  vegetables,  the  eye  is  delighted  with  the  regularity 
and  beauty  which  characterizes  the  texture  and  intricate  struc- 
ture of  plants  and  flowers.  The  most  delicate  fibres  of  a  leaf, 
the  pores  through  which  the  vegetable  fluids  circulate,  the  downy 
covering  of  plants  and  foliage,  as  of  certain  mosses,  which  is  too 
minute  to  disclose  its  figure  to  the  naked  eye, — objects  of  this 
kind,  when  expanded  under  the  solar  microscope,  astonish  and 
delight  us  by  the  symmetry  of  their  structure.  Their  appropri- 
ate colors  are  not  so  well  exhibited  by  this  instrument,  as  by 
some  other  forms  of  the  microscope  to  be  described  hereafter. 
In  the  animal  kingdom,  the  solar  microscope  extends  the  range 
of  vision  in  a  manner  no  less  surprising  and  instructive.  The 
minutest  insects  we  are  acquainted  with,  are  exhibited  to  us  as 
animals  of  the  largest  size,  and  often  of  monstrous  shapes,  from 
the  multiplicity  of  their  parts  and  apparent  disproportion  ;  and 
animalcules,  or  those  members  of  the  animal  creation  which  are 
too  minute  to  be  seen  at  all  by  the  naked  eye,  are  suddenly 
brought  into  life  in  countless  numbers.  The  forms,  the  motions, 
and  the  habits  of  these  beings,  are  among  the  most  curious  reve- 
lations of  the  solar  microscope.  The  circulation  of  the  blood  may 
je  seen  in  the  fins  of  fishes  and  other  transparent  parts  of  ani- 
mals, presenting  a  very  curious  and  interesting  spectacle.  The 
•rystallization  of  salts,  which  may  be  exhibited  while  the  crystals 


572 


NATURAL    PHILOSOPHY. 


are  forming  and  arranging  themselves,  (as  many  of  them  do 
with  great  precision  and  symmetry,)  is  among  the  finest  repre- 
sentations of  this  instrument. 

Since  the  light  is  transmitted  through  the  object,  it  will  of 
course  be  understood,  that  only  such  objects  as  are  transparent 
can  be  employed  in  the  manner  already  described.  In  some  va- 
rieties of  the  solar  microscope,  there  are  special  contrivances 
for  exhibiting  opake  objects  by  means  of  reflected  light. 

848.  If  we  form  an  image  of  an  object  with  Fig.  296. 

the  single  microscope,  (as  is  done  in  the  magic 
lantern  and  solar  microscope,)  when  that  im- 
age is  not  too  large,  we  may  obviously  apply 
to  it  a  magnifier,  as  we  would  to  an  original 
object  of  the  same  size.  This  is  the  principle 
of  the  compound  microscope. 

The  COMPOUND  MICROSCOPE  consists  of  at  least 
two  convex  lenses,  one  of  which,  called  the 
object-glass,  is  used  to  form  an  enlarged  image 
of  the  object,  and  the  other,  called  the  eye- 
glass,  is  used  to  magnify  the  image  still  fur- 
ther. 

Thus,  let  ab,  (Fig.  296,)  be  the  object,  be- 
ing placed  a  little  further  from  the  object- 
glass,  cd,  than  the  principal  focus,  then  the 
rays  of  light  emanating  from  it  will  be  col- 
lected on  the  other  side  of  the  lens  and  form  an  image,  gh, 
whose  diameter  is  as  much  larger  than  that  of  the  object  as  its 
distance  from  the  lens  is  greater.  (Art.  760.)  Let  ef  be  the 
eye-glass,  which  must  be  placed  at  such  a  distance  from  the  im- 
age, that  the  latter  shall  be  in  the  focus  of  parallel  rays  ;  then 
the  rays  proceeding  from  the  image  will  go  out  parallel,*  and 
come  to  the  eye,  situated  behind  the  glass,  in  a  state  favorable 
for  distinct  vision. 


849.  The  magnifying  power  of  the  Compound  Microscope  is 
estimated  as  follows.  First,  the  diameter  of  the  image  will  be 
to  that  of  the  Abject  as  their  respective  distances  from  the  lens. 
Secondly,  the  image  is  magnified  by  the  eye-glass  according  to 
the  principles  of  the  single  microscope,  (Art.  376,)  namely,  in  the 
ratio  of  its  focal  distance  to  the  limit  of  distinct  vision.  Thus, 
suppose  the  image  is  formed  at  ten  times  the  distance  of  the  ob- 
ject ;  it  will  of  course  be  magnified  ten  times.  Again,  suppose 

*  It  is  to  be  remarked  here,  and  in  all  similar  cases,  that  it  is  only  the  rays  of  each 
individual  pencil  that  are  parallel ;  that  is,  those  rays  which  come  from  the  same 
point  in  the  object.  The  rays  of  different  pencils  may  cross  each  other  variously,  and 
the  different  pencils  may  converge  or  diverge  among  themselves ;  still,  if  the  rays  of 
each  pencil  are  parallel  to  one  another,  the  vision  will  be  distinct. 


OPTICS.  573 

the  eye-glass  has  a  focal  distance  of  one  inch,  the  limit  of  dis- 
tinct vision  being  five  inches ;  the  image  will  be  further  magni- 
fied five  times ;  by  both  glasses,  therefore,  the  object  will  be  mag- 
nified fifty  times.  If  the  first  ratio  be  that  of  one  to  one  hundred, 
then  the  instrument  will  magnify  the  linear  dimensions  five  hun- 
dred times,  and  the  surface  two  hundred  and  fifty  thousand  times. 
From  this  double  magnifying  process,  it  might  be  supposed  that, 
by  means  of  the  compound  microscope,  it  would  be  easy  to  at- 
tain a  much  higher  magnifying  power  than  by  the  single  micro- 
scope ;  but  this  is  not  the  fact,  for,  in  the  first  place,  we  cannot 
form  an  image  of  a  size  beyond  certain  moderate  limits,  without 
making  it  too  large  for  the  eye-glass  to  cover  ;  or,  if  an  eye-glass 
of  very  large  field  of  view  be  employed,  its  focal  distance  must 
be  great,  and  consequently  its  magnifying  power  small.  We  are, 
therefore,  unable  to  employ  so  high  a  magnifier  for  our  object- 
glass  as  we  may  apply  to  the  naked  eye,  and  we  can  employ  only 
a  microscope  of  still  inferior  power  for  our  eye-glass. 

850.  On  account  of  the  necessity  of  using  a  large  eye-glass  to 
view  the'  magnified  image,  compound  microscopes  require  to  have 
the  tube  which  contains  the  glasses  larger  toward  the  eye-glass 
than  toward  the  object-glass.  Sometimes  the  magnifiers  are 
contained  in  a  box  of  pyramidal  shape,  the  reason  of  which  is 
obvious.  Of  the  latter  figure  is  the  Lucernal  Microscope,  a  va- 
riety of  the  compound  microscope,  which  admits  of  being  used 
with  the  light  of  a  lamp  instead  of  day  light,  and  is  furnished 
with  a  reflector  and  a  condensing  lens,  by  one  or  the  other  of 
which  the  light  of  the  lamp  may  be  concentrated  upon  the  object. 
The  lucernal  microscope  is  furnished  with  a  piece  of  ground 
glass,  upon  which  the  image  may  be  received  as  upon  a  screen. 
The  object  being  illuminated  by  a  lamp,  and  the  image  being 
seen  in  a  dark  room,  this  arrangement  is  very  convenient  for 
drawing  insects,  flowers,  &c.  Although  the  compound  does  not 
possess  higher  magnifying  powers  than  the  simple  microscope, 
yet  it  commands  a  much  greater  field  of  view.  We  view  the 
image  with  the  eye-glass  in  the  same  manner  as  we  view  the  ob- 
ject with  a  single  microscope  ;  but  having  already  a  magnified 
representation  of  the  object,  we  have  no  occasion  to  apply  to  the 
eye  so  high  a  magnifier,  and  therefore  we  may  employ  one  of 
greater  focal  distance,  which  consequently  takes  in  a  greater  field 
of  view.  The  field  of  view  is  still  further  improved  in  some 
compound  microscopes  by  interposing  a  field-glass,  which  is  a 
convex  lens,  introduced  between  the  eye-glass  and  the  place  of 
the  image,  and  near  the  latter,  (as  a  little  above  gh,  Fig.  296,) 
the  effect  of  which  is  to  diminish  the  divergency  of  the  pencils 
of  rays,  and  thus  to  bring  into  the  range  of  the  eye-glass 
those  pencils,  which  would  otherwise  diverge  too  much  to 


674  NATURAL    PHILOSOPHY. 

fall  within  it.  It  has  been  before  remarked,  that  the  cornea  per- 
forms a  similar  office  for  the  crystalline  lens  of  the  eye.  (Art. 
823.) 

851.  Instead  of  employing  a  convex  lens  for  the  purpose  of 
forming  an  image  of  the  object,  we  may  use  a  concave  mirror  for 
the  same  purpose.     On  this  principle  are  constructed  REFLECTING 
MICROSCOPES.     The  object  being  placed  before  the  mirror,  at  a 
distance  a  little  greater  than  the  focal  distance,  a  magnified  image 
will  be  formed  on  the  other  side  of  the  center,  as  in  Fig.  256.    To 
this  image  we  may  obviously  apply  an  eye-glass,  in  the  same 
manner  as  in  the  common  compound  microscope.     Reflecting  mi 
croscopes  are  supposed  to  have  some  advantage  over  the  refract- 
ing, but  they  have  not  come  into  general  use.    By  making  the  con- 
cave reflector  of  a  parabolic  figure,  spherical  aberration  is  pr»e- 
vented,  and  reflectors  are  not  liable,  like  lenses,  to  form  colored 
images  in  consequence  of  the  decomposition  of  the  light  into  its 
prismatic  rays,  called  chromatic  aberration.     These  difficulties, 
however,  when  they  occur,  admit  of  being  obviated  by  peculiar 
contrivances,  which  will  be  more  particularly  described  in  con- 
nection with  telescopes. 

852.  Dr.  Brewster  gives  the  following  rules  for  making  mi- 
croscopic observations. 

1.  The  eye  should  be  protected  from  all  extraneous  light,  and 
should  not  receive  any  of  the  light  which  proceeds  from  the  illu- 
minating body,  excepting  what  is  transmitted  through,  or  is  re- 
flected from,  the  object. 

2.  Delicate  observations  should  not  be  made  when  the  fluid 
which  lubricates  the  cornea  is  in  a  viscid  state. 

3.  The  best  position  for  microscopical  observations,  is  when 
the  observer  is   lying  horizontally  on   his  back.      This  arises 
from  the  perfect  stability  of  his  head,  and  from  the  equality  of 
the  lubricating  film  of   fluid  which  covers  the  cornea.      The 
worst  of  all  positions  is  that  in  which  we  look  downward  ver- 
tically. 

4.  If  we  stand  straight  up  and  look  horizontally,  parallel  mark- 
ings or  lines  will  be  seen  most  perfectly  when  their  direction  is 
vertical ;  viz.  the  direction  in  which  the  lubricating  fluid  descends 
over  the  cornea. 

5.  Every  part  of  the  object  should  be  excluded  except  that 
which  is  under  immediate  observation. 

6.  The  light  which  illuminates  the  object,  should  have  a  very 
small  diameter.     In  the  daytime  it  should  be  a  single  hole  in  the 
window  shutter  of  a  darkened  room,  and  at  night  an  aperture 
placed  before  an  argand  lamp. 

7.  In  all  cases,  particularly  when  high  powers  are  used,  the 


OPTICS. 


575 


297. 


natural  diameter  of  the  illuminating  light  should  be  diminished, 
and  its  intensity  increased,  by  optical  contrivances.* 

The  microscope  is  sometimes  employed  to  form  images  for  the 
purposes  of  drawing.  In  this  manner  landscapes  are  represent- 
ed, objects  of  natural  history  are  delineated,  and  artificial  pic- 
tures are  reduced  and  copied.  The  two  instruments  particularly 
employed  for  this  purpose,  are  the  Portable  Camera  Obscura  and 
the  Camera  Lucida. 

853.  The  PORTABLE  CAMERA  OBSCURA,  which  is  used  for  delin- 
eating landscapes,  and  much  more  of  late  for  taking  likenesses  by 
the  Daguerreotype  process,  consists  of  a  wooden  box,  (answering 
to  the  dark  chamber,  Art.  818,)  with  which  is  connected  a  convex 
lens,  so  exposed  to  the  landscape  as  to  receive  the  rays  of  light 
from  the  various  objects  in  it,  and  form  a  picture  of  them  on  a 
screen  placed  within  the  box  at  the  focal  distance  of  the  lens. 
Such  is  a  general  description  of  the  instrument,  of  which  there 
are  several  different  forms.  The  following  diagram  represents 
a  common  convenient  form. 

ABCD,  (Fig.  297,)  a  box  usually  made 
of  thin  pieces  of  mahogany. 

ad,  a  plano-convex  lens,  this  form  being 
preferred  because  it  has  less  aberration 
than  a  double  convex.  (Art.  "762.) 

ED,  a  plane  mirror,  turning  on  a  hinge 
at  D,  and  capable  of  being  raised  or  low- 
ered, so  as  to  admit  more  or  less  of  the 
landscape. 

be,  a  piece  of  pasteboard,  covered  with 
a  sheet  of  fine  white  paper,  and  bent  so 
as  to  form  a  concave  screen,  and  placed 
at  the  focal  distance  of  the  lens.  A  cast- 
ing of  stucco,  of  the  figure  of  a  concave 
portion  of  a  sphere,  affords  the  most  per- 
fect picture.  A  B 

The  rays  of  light  from  external  objects,  falling  upon  the  mir- 
ror ED,  are  conveyed  to  the  lens  in  the  same  manner  as  though 
they  came  directly  from  objects  at  the  same  distance  behind  the 
mirror.  Passing  through  the  lens,  they  are  brought  to  a  focus, 
and  form  a  picture  of  the  landscape  on  the  screen,  which  may 
be  viewed  by  an  opening  in  the  side  of  the  box  at  F,  and  may 
be  copied  by  a  hand  introduced  into  the  box  by  an  opening  be- 
low. 

Although  the  image  is  inverted  with  respect  to  the  objects,  yet 
as  the  spectator,  in  looking  into  the  box,  stands  with  his  back  to 
the  landscape,  the  picture  appears  erect. 

*  Brewster's  Optics,  345. 


576 


NATURAL  PHILOSOPHY. 


854.  The  CAMERA  LUCIDA  is  an  instrument  of  more  recent  ori- 
gin, having  been  invented  by  the  late  Dr.  Wollaston.  It  consists 
of  a  prism  so  contrived  that  its  surfaces,  by  their  reflecting  prop- 
erties,* give  the  proper  direction  to  the  rays  of  light,  and  finally 
project  an  image  of  the  object  in  a  convenient  position  for  copy- 
ing, as  is  represented  in  the  following  diagram. 


Fig.  298. 


-4-M 


ABCD,  (Fig.  298,)  is  a 
glass  prism,  having  the  an- 
gle at  A  90°,  the  angle  at 
D  67£°,  the  angle  at  C 
135°.  In  taking  an  obser- 
vation the  prism  is  set  with 
the  side  AD  parallel  to  the 
object  M.  A  ray  of  light 
ND,  falling  perpendicular- 
ly upon  AD,  suffers  no  re- 
fraction, bwt  proceeds  on  to 
the  second  surface  DC, 
where  it  makes  with  DC 
an  angle  of  22£°,  (the  com- 

Slement  of  the  angle  at  D.) 
f  course  the  angle  CGH  is  221°,  and  these  two  angles,  sub- 
tracted from  180°,  leave  NGH=135°.  Again,  since  GCH=135°, 
and  CGH=22!°,  therefore  CHG  a*id  BHE  each  equal  22i°,  and 
therefore  GHE=135°.  Produce  NG  till  it  meets  HM'  in  I ;  then 
the  angles  1GH  and  IHG  will  be  severally  45°,  and  consequently 
HIG  (which  is  the  angle  made  by  the  incident  and  emergent 
rays)  will  be  90°.  Therefore,  the  perpendicular  object,  MN,  will 
appear  to  the  eye  on  a  horizontal  plane  at  M',  as  far  behind  the 
reflecting  surface  as  M  is  before  it.  (Art.  729.)  Now  if  the 
prism  is  so  formed,  that  the  emergent  rays  shall  be  very  near  the 
angular  point  B,  the  eye  situated  at  E  may  take  in  at  once  the 
image  and  the  paper  on  which  it  is  projected,  seeing  the  former 
through  the  prism  and  the  latter  by  direct  vision ;  and  thus  the 
image  may  be  very  perfectly  sketched.  This  beautiful  instru- 
ment is  usually  mounted  in  a  case,  and  has  various  appendages, 
which  severally  contribute  to  its  utility,  but  we  aim  only  to  con- 
vey an  idea  of  its  principle.^ 

*  It  will  be  observed,  in  the  following  illustration,  that  the  rays  of  light  strike  the 
surfaces  of  the  prism  at  such  an  angle  as  to  undergo  total  reflexion.  (Art.  748.) 

t  For  a  more  extended  description  of  the  Camera  Lucida,  see  Nicholson's  Phil 
Journal,  and  Tilloch's  Phil.  Magazine,  for  1807. 


OPTICS. 


577 


CHAPTER  X. 


OF    TELESCOPES. 

855.  THE  Telescope  is  an  optical  instrument,  designed  to  aid  the 
eye  in  viewing  distant  objects.* 

The  construction  of  this  noblest  of  instruments,  in  its  different 
forms,  involves  the  application  of  all  the  leading  principles  of  the 
science  of  Optics.  The  study  of  the  Telescope  is  therefore  the 
study  of  the  science,  and  a  distinct  enunciation  of  the  principles 
involved  in  it,  will  serve  as  a  recapitulation  of  the  most  useful 
principles  of  Optics.  The  advantage  which  the  student  will  de- 
rive from  reviewing  these  points,  as  exemplified  in  their  appli- 
cation, will  justify  us  in  bringing  up  distinctly  to  view  various 
principles  already  unfolded. 

856.  The  leading  principle  of  the  Telescope  may  be  thus 
enunciated : 

By  means  of  either  a  convex  lens,  or  a  concave  mirror,  an  image 
of  the  object  is  formed,  which  is  viewed  and  magnified  with  a  micro- 
scope. 

Thus,  let  ABCD  represent  the  tube  of  a  telescope.  At  the 
front  end,  or  the  end  which  is  directed  toward  the  object,  (which 
we  will  suppose  to  be  the  moon,)  is  inserted  a  convex  lens  L, 
which  receives  the  rays  of  light  from  the  moon,  and  collects 
them  into  the  focus  a,  forming  an  image  of  the  moon.  This 
image  is  viewed  by  a  magnifier  attached  to  the  end  BC. 
Fig.  299. 


The  most  general  division  of  the  instrument  is  into  Refracting 
and  Reflecting  Telescopes :  of  which  the  former  produce  their 
image  by  means  of  a  convex  lens,  and  the  latter  by  means  of  a 
concave  mirror.  The  instrument,  according  to  the  uses  to  which 
it  is  applied,  receives  the  farther  denominations  of  the  Astro- 
nomical and  the  Terrestrial  Telescope  ;  and  also  telescopes  are 


t,  at  a  distance, 

73 


,  to  see 


578  NATURAL   PHILOSOPHY. 

named  after  their  several  inventors,  Galileo's,  Newton's,  Grego- 
ry's, Herschel's,  &c. 


THE    ASTRONOMICAL   TELESCOPE. 


857.  We  begin  with  this  variety  because  it  is  one  of  the  most 
simple,  and  because,  in  connection  with  it,  we  may  conveniently 
study  the  theory  of  the  instrument  at  large. 

The  Astronomical  Telescope  has  essentially  but  two  glasses : 
these  are  usually  fixed  in  a  tube  of  brass,  one  at  one  end,  and  the 
other  at  the  other  end.  The  glass  at  the  end  of  the  tube  which 
is  directed  to  the  object,  is  called  the  object-glass ;  and  that  at 
the  end  to  which  the  eye  is  applied,  is  called  the  eye-glass.  The 
object-glass  is  a  convex  lens  which  forms  an  image  of  a  distant 
object,  as  a  star,  in  its  focus  of  parallel  rays,  and  the  eye-glass 
is  a  microscope  with  which  we  view  the  image,  at  a  distance 
equal  to  its  focus  of  parallel  rays.  Of  course,  the  distance  of  the 
two  glasses  from  each  other  is  equal  to  the  sum  of  their  focal 
distances.  See  the  annexed  figure. 

Fig.  300. 
A' 
B' 


MN,  object-glass. 
PQ,  eye-glass. 

A'D',  AD,  A"D",  parallel  rays  from  the  top  of  the  object. 
B'D',  BD,  B"D",        "          "       "  center  do. 

CD',  CD,  C"D",         "         "       "  bottom  do. 

ba,  inverted  image  formed  in  the  focus  of  parallel  rays. 
&PF,  a  pencil  of  rays,  proceeding  from  the  top  of  the  image 
to  the  eye-glass,  and  rendered  parallel. 
cKF,  a  similar  pencil  from  the  center. 
aQP,         do.  from  the  bottom. 

F,  point  where  the  different  pencils  cross  the  axis. 

858.  In  this  instrument  we  observe  a  striking  resemblance  to 
the  Compound  Microscope.  (Fig.  296.)  In  the  microscope, 
however,  since  the  object  is  nearer  than  the  image,  the  image  is 
greater  than  the  object ;  but  in  the  telescope,  since  the  object  is 
removed  to  a  great  distance,  the  image  is  formed  much  nearer 
to  the  lens  than  the  object,  and  is  proportionally  smaller.  Hence, 
compound  microscopes  have  their  tubes  enlarged  in  diameter  to- 
ward the  eye-glass,  while  telescopes  have  their  tubes  diminished 


OPTICS.  579 

hi  that  direction.  Since  the  vertical  angles  at  D,  subtended  on 
the  one  side  by  the  object,  and  on  the  other  by  the  image,  are 
equal,  were  the  eye  situated  at  the  center  of  the  object-glass,  it 
would  see  the  object  and  the  image  under  the  same  visual  angle, 
and,  consequently,  both  would  appear  of  the  same  magnitude. 
Moreover,  were  the  eye  placed  at  the  same  distance  from  the 
image  on  the  other  side  of  it,  it  would  be  apparently  of  the  same 
size  as  before,  and  therefore  of  the  same  apparent  diameter  as 
the  object.  But  by  means  of  a  microscope,  such  as  the  eye-glass 
in  fact  is,  we  may  view  it  at  a  much  nearer  distance,  and  of 
course  magnify  it  to  any  extent,  as  was  fully  shown  in  explain- 
ing the  principles  of  the  simple  microscope.  (Art.  837.)  Hence 
the  magnifying  power  of  the  telescope  depends  on  the  ratio  be- 
tween the  focal  distances  of  the  object-glass  and  the  eye-glass. 
If,  as  in  the  figure,  the  common  focus  is  ten  times  nearer  the  eye- 
glass than  the  object-glass,  the  instrument  will  magnify  ten 
times  ;  if  one  hundred  times  nearer,  one  hundred  times  ;  and  so 
in  all  other  cases.  Hence  we  may  increase  the  magnifying  pow- 
er of  the  instrument,  either  by  employing  an  object-glass  of  very 
small  curvature,  which  throws  its  image  to  a  great  distance,  or 
an  eye-glass  of  high  curvature  and  small  focal  distance.  Sup- 
pose, for  example,  the  object-glass  has  a  focal  distance  of  forty 
feet,  or  four  hundred  and  eighty  inches,  and  the  eye-glass  has  a 
focal  distance  of  one  tenth  of  an  inch,  then  the  magnifying  pow- 
er of  this  instrument  would  be  four  thousand  and  eight  hundred 
in  diameter,  and  the  square  of  this  number  in  surface. 

859.  As  the  sphericity  of  the  eye-glass  may  be  increased  in- 
definitely, and  its  focal  distance  diminished  to  the  same  extent, 
it  would  seem  possible  to  apply  very  high  magnifying  powers  in 
very  short  telescopes.  For  example,  suppose  the  focal  distance 
of  the  object-glass  is  twenty-four  inches  ;  by  using  a  microscope 
of  TVth  of  an  inch  focus,  we  have  a  power  of  two  hundred  and 
forty.  But  it  must  be  kept  in  mind,  that  such  microscopes  com- 
mand only  an  exceedingly  small  field  of  view,  and  would,  there- 
fore, not  enable  us  to  see  any  thing  more  than  a  minute  portion 
of  an  object  of  any  considerable  size  ;  and  not  sufficient  light 
would  be  transmitted  through  such  an  aperture  to  answer  the 
purposes  of  vision.  Since  the  image  is  inverted  with  respect  to 
the  object,  and  is  viewed  in  this  situation  by  the  glass,  objects 
seen  through  Astronomical  Telescopes  appear  inverted.  By  the 
addition  of  several  more  lenses,  they  may  be  made  to  appear 
erect,  as  will  be  shown  in  the  description  of  the  Day  Glass,  or 
Terrestrial  Telescope ;  but  at  every  new  refraction  a  certain 
portion  of  light  is  extinguished,  a  loss  which  it  is  important  to 
avoid  in  instruments  designed  to  be  used  at  night ;  while,  in 
regard  to  celestial  objects,  it  is  not  essential  whether  they  are 
seen  erect  or  inverted. 


580  NATURAL   PHILOSOPHY. 

The  place  for  the  eye  to  view  the  image  with  the  best  advan- 
tage is  at  F,  where  the  pencils  of  parallel  rays  meet. 

860.  The  difficulties  to  be  overcome  in  the  construction  of  a 
perfect  Refracting  Telescope,  (some  of  which  are  very  formida- 
ble,) are  chiefly  the  following  :    1.    Spherical  aberration ;    2. 
Chromatic  aberration  ;  3.  Want  of  sufficient  light ;  4.  Want  of 
a  field  of  view  sufficiently  ample  ;    5.  Imperfections  of  glass. 
Each  of  these  particulars  we  will  briefly  consider. 

861.  Spherical  Aberration,  it  will  be  recollected,  occasions  in- 
distinctness in  images  formed  by  lenses,  in  consequence  of  the 
different  rays  of  the  same  pencil  not  being  all  brought  to  a  focus 
at  the  same  point,  those  which  fall  upon  the  extreme  parts  of 
the  lens  being  more  refracted  and  coming  to  a  focus  sooner  than 
those  which   are  nearer  to  the  axis.      (See   Art.  761.)      The 
amount  of  this  error  is  found  to  depend  on  two  circumstances, 
namely,  the  diameter  of  the  lens,  or  what  is  technically  called 
its  aperture*  and  its  focal  distance,  increasing  rapidly  as  the 
aperture  is  increased,  and  diminishing  as  the  focal  distance  is 
increased.!     Small  apertures  and  flat  or  thin  lenses  are,  therefore, 
most  free  from  spherical  aberration.     But  if  we  use  smajl  aper- 
tures we  cannot  have  a  strong  light,  which  is  a  circumstance  of 
the  greatest  importance  in  astronomical  observations,  since  it  is 
of  little  consequence  to  enlarge  the  dimensions  of  an  object  if 
we  have  not  light  enough  to  render  it  visible.     Indeed,  many 
astronomical  objects,  as  small  stars,  are  rendered  visible  by  the 
telescope,  not  in  consequence  of  any  apparent  increase  of  size, 
but  because  this  instrument  collects  and  conveys  to  the  eye  a 
much  larger  beam  of  light  from  them  than  would  otherwise  en- 
ter it.     While  the  diameter  of  the  beam  which  falls  upon  the 
naked  eye  is  only  the  fraction  of  an  inch,  that  collected  by  the 
telescope  may  be  several  inches,  or  even  several  feet,  according 
to  the  size  of  the  instrument.     Hence  the  advantage  of  large 
apertures  is  obvious.     Again,  we  cannot  wholly  remedy  the 
error  in  question,  though  we  may  diminish  it  by  using  very  flat 
lenses  which  have  great  focal  distances ;  but  the  tendency  of 
this  expedient  is  to  render  the  instrument  inconveniently  long. 
Other  expedients,  therefore,  become    necessary  for    correcting 
spherical  aberration  in  refracting  telescopes. 

*  The  aperture,  strictly  speaking,  is  the  diameter  of  that  part  of  the  lens  through 
which,  in  a  given  case,  light  is  admitted,  whether  it  be  the  whole  surface  or  only  & 
part  of  it. 

t  It  is  found  by  opticians,  that  the  longitudinal  aberration  of  lenses  increases  as 
the  square  of  the  aperture,  with  a  given  curvature,  and  is  inversely  as  the  focal  dis- 
tance, with  a  given  aperture,  and  that  the  lateral  aberration  increases  as  the  cube  of 
the  aperture,  with  a  given  radius,  or  inversely  as  the  square  of  the  radius  with  a  given 
aperture. 


OPTICS.  581 

862.  In  the  eye-glasses,  which  are  liable  to  the  same  difficulty, 
where  the  lens  has  a  great  curvature,  as  is  the  case  with  such 
as  have  high  magnifying  powers,  the  aperture  is  necessarily  re- 
duced very  much,  by  excluding  all  the  light  except  what  passes 
through  the  central  parts  of  the  lens.  At  least  this  is  the  case 
where  glass  lenses  are  used.  But  the  microscopes  made  of  dia- 
mond, sapphire,  and  other  gems,  have  not  only  high  refractive 
powers,  but  are  less  subject  to  spherical  aberration  than  similar 
lenses  of  glass.  (Art.  839.)  Thus,  if  three  lenses  were  ground 
in  the  same  tool,  one  of  plate  glass,  one  of  sapphire,  and  the 
other  of  diamond,  their  respective  magnifying  powers  and  aber- 
rations would  be  as  follows  :* 

Magnifying  power.  Longitudinal  aberration. 

Glass, 150     ......     1.167 

Sapphire, 250 1.005 

Diamond 400 0.950 

This  difference  in  aberration  will  be  much  greater  if  the  lenses 
be  so  formed  as  to  give  the  same  magnifying  powers ;  for  then 
the  diamond  and  sapphire  lenses  may  be  made  so  much  thinner 
as  greatly  to  reduce  the  aberration. 

But  although  eye-pieces,  on  account  of  their  small  size,  may 
sometimes  be  made  of  the  precious  gems,  yet  this  can  rarely  be 
the  case  on  account  of  the  great  expense  attending  them.  It  is 
obvious,  also,  that  they  cannot  be  employed  for  the  object  lenses. 
The  most  successful  method  of  diminishing  spherical  aberration 
in  eye-pieces  of  glass,  is  by  a  combination  of  plano-convex  lenses, 
by  means  of  which  a  given  refracting  power  may  be  attained 
with  far  greater  distinctness  than  by  a  single  lens  of  the  same 
power.  Thus,  when  two  piano-  Fig.  301. 

convex  lenses  are  placed  as  in 
Fig.  301,  it  is  found  that  the  im- 
age has  four  times  the  distinct-  ...- 
ness  of  a  double  convex  lens  of 
equivalent  power. f  Here  F  is  a 
lens  which  would  bring  the  par- 
allel rays  to  a  focus  and  form  the  image  at  the  distance  of  G  ; 
but  E  is  another  similar  lens,  which,  receiving  them  in  a  con- 
verging state,  makes  them  converge  more,  and  come  to  a  focus 
at  H.  The  double  convex  lens  D,  would  do  the  same,  but  with 
much  greater  spherical  aberration.  It  appears,  indeed,  that  the 
spherical  aberration  may  be  wholly  removed  by  combining  a 
meniscus  with  a  double  convex  lens  of  certain  curvatures.  J 

*  The  figure  of  the  lens  is  supposed  to  be  plano-convex,  the  convex  side  being 
turned  toward  parallel  rays. 

t  The  Scioptfc  Ball,  used  in  the  camera  obscura,  (Art.  819,)  is  formed  of  two  such 
lenses. 

t  See  Brewster's  Optics,  p.  58,  or  Herschel  on  Light,  Sec.  316. 


582  NATURAL    PHILOSOPHY. 

863.  In  object-glasses,  which,  on  account  of  their  smaller  cur- 
vatures, are  not  so  subject  to  error  from  spherical  aberrations  as 
eye-glasses  are,  the  most  advantageous  form  is  that  of  a  double 
convex  lens  of  unequal  curvatures,  the  radii  of  the  opposite  sur- 
faces being  as  one  to  six,  (Art.  764,)  and  the  less  convex  side 
being  turned  toward  the  parallel  rays. 

In  short,  it  appears,  that  in  order  to  avoid  the  errors  arising 
from  spherical  aberration  in  large  lenses,  they  must  be  made  as 
thin  as  convenience  will  permit ;  that  where  it  is  practicable, 
they  may  be  most  advantageously  formed  of  the  precious  gems, 
particularly  the  diamond ;  that  a  plano-convex  lens  with  its  con- 
vex side  toward  the  parallel  rays  has  less  aberration  than  a 
double  convex  lens  of  equivalent  power  ;  that  two  plano-convex 
lenses  may  be  so  combined  as  to  have  only  £  as  much  aberration 
as  the  double  lens,  and  a  meniscus  may  be  so  united  to  a  double 
convex  lens  as  wholly  to  prevent  aberration ;  and  finally,  that 
the  aberration  may  be  reduced  to  a  very  small  error  simply  by 
employing  a  double  convex  lens  whose  curvatures  on  the  oppo- 
site sides  are  as  one  to  six. 

Since  lenses  having  the  curvature  of  one  of  the  conic  sections 
are  free  from  spherical  aberration,  Sir  Isaac  Newton  ground  an 
object-glass  into  the  figure  of  a  paraboloid.  This  was  free  from 
the  error  in  question,  but  involved  another  still  more  formidable, 
since  it  decomposed  the  light  and  gave  an  image  tinged  with  the 
colors  of  the  rainbow.  On  observing  this,  Sir  Isaac  pronounced 
the  further  improvement  of  the  refracting  telescope  to  be  hope- 
less, and  betook  himself  to  exclusive  efforts  for  improving  the 
reflecting  telescope.  But  the  combined  ingenuity  of  philosophers 
and  artists,  has  nearly  overcome  this  error  also. 

864.  The  next  difficulty  therefore  to  be  considered  is  that 
which  arises  from  the  separation  of  the  prismatic  colors,  in  con- 
sequence of  the  different  refrangibility  of  the  different  rays,  an 
error  which  is  called  Chromatic  Aberration. 

The  general  principles  of  Chromatic  Aberration,  will  be  readi- 
ly comprehended  by  calling  to  mind,  that  distinct  images  are 
formed  only  when  the  rays  of  the  same  pencil  which  flow  from 
any  point  in  the  object  are  collected  into  one  and  the  same  point 
in  the  image,  unmixed  with  rays  from  any  other  point ;  that  the 
prismatic  rays  which  compose  white  light  have  severally  differ- 
ent degrees  of  refrangibility,  some  being  more  turned  out  of  their 
course  than  others,  in  passing  through  the  same  medium  ;  that, 
consequently,  the  different  colored  rays  of  the  same  pencil  would 
meet  in  different  points,  each  set  of  colored  rays  forming  its  own 
image,  but  all  these  images  becoming  blended  with  one  another, 
and  thus  composing  a  confused,  colored  picture. 

To  illustrate  these  principles  let  LL  be  a  lens  of  crown  glass, 
and  RL,  RL,  rays  of  white  light  incident  upon  it,  parallel  to  its 


OPTICS.  583 

axis  Rr.     Let  the  extreme  Fig.  302. 

violet  rays,  whose  index  of 

refraction  is  1.54666,  be  re-  R" 

fracted  so  as  to  meet  the  « 

axis  in  v  ;  then  the  extreme 

red,  whose  index  of  refrac-  R. 

tion    is    only    1.5258,  will 

meet  the  axis  at  some  point  more  distant  from  the  lens,  as  at  r. 

Cv  and  Cr  are  the  focal  distances  of  the  lens  for  the  violet  and 

the  red  rays  respectively.      The  distance  vr  is  the  chromatic 

aberration,  and  the  circle  whose  diameter  is  ab,  which  passes 

through  the  focus  of  the  mean  refrangible  rays  at  o,  is  called  the 

circle  of  least  aberration. 

865.  These  effects  may  be  shown  experimentally  by  exposing 
the  lens  LL,  (Fig.  302,)  to  the  parallel  rays  of  the  sun.  If  we 
receive  the  image  of  the  sun  on  a  piece  of  paper  placed  between 
o  and  C,  the  luminous  circle  on  the  paper  will  have  a  red  border, 
becaure  it  is  a  section  of  the  cone  LrL,  the  exterior  rays  of 
which  La,  Lb,  are  red  ;  but  if  the  paper  is  placed  at  any  greater 
distance  than  o,  the  luminous  circle  on  the  paper  will  have  a 
violet  border,  because  it  is  a  section  of  the  cone  Iv'l',  the  exterior 
rays  of  which  al',  bl'  are  violet,  being  a  continuation  of  the  violet 
rays  Lv,  Lv.  As  the  spherical  aberration  of  the  lens  is  here 
combined  with  its  chromatic  aberration,  the  undisguised  effect 
of  the  latter  will  be  better  seen  by  taking  a  large  convex  lens 
LL,  and  covering  up  all  the  central  part,  leaving  only  a  small 
rim  round  its  circumference  at  LL,  through  which  the  rays  of 
light  may  pass.  The  refraction  of  the  differently  colored  rays 
will  then  be  finely  displayed  by  viewing  the  .image  of  the  sun 
on  the  different  sides  of  ab. 

It  is  clear  from  these  observations,  that  the  lens  will  form  a 
violet  image  of  the  sun  at  v,  a  red  image  at  r,  and  images  of  Jhe , 
other  colors  of  the  spectrum  at  intermediate  points  between  r 
and  v  ;  so  that  if  we  place  the  eye  behind  these  images,  we  shall 
see  a  confused  image,  possessing  none  of  that  sharpness  and  dis- 
tinctness which  it  would  have  had  if  formed  only  by  one  kind 
of  rays.* 

The  separation  of  white  light  into  its  prismatic  colors,  is  called 
dispersion  ;  and  the  comparative  power  of  effecting  this  separa- 
tion, possessed  by  different  media,  is  called  the  dispersive  power. 
The  dispersive  power  is  measured  by  the  ratio  which,  in  any  case, 
the  separation  of  the  red  and  violet  rays  bears  to  the  mean  re- 
fraction of  the  compound  ray.  Thus  if  a  ray  of  solar  light  on 
passing  through  a  lens,  is  turned  out  of  its  original  direction  27°, 
and  the  red  and  violet  rays  are  separated  from  each  other  1°, 

*  Brewster's  Optics,  p.  79 


581  NATURAL  PHILOSOPHY. 

then  the  dispersive  power  is  said  to  be  ?\,  which  is  usually  ex- 
pressed in  the  form  of  a  decimal  fraction,  .037=^. 

866.  Different  bodies  possess  different  dispersive  powers. 

The  dispersive  powers  of  a  few  of  the  most  important  sub- 
stances in  relation  to  the  subject  before  us,  are  exhibited  in  the 
following  table. 

Dispersive  power.  Dispersive  power. 

Oil  of  Cassia,       .     .     .     0.139  Plate  Glass,       .  .     0.032 

Sulphuret  of  Carbon,     .  0.130  Sulphuric  Acid,  .  .  0.031 

Oil  of  Bitter  Almonds,      0.079  Alcohol,    ....     0.029 

Flint  Glass,       ....  0.052  Rock  Crystal,      .  .  0.026 

Muriatic  Acid,     .     .     .     0.043  Blue  Sapphire,  .     0.026 

Diamond,      .     ....     .  0.038  Fluor  Spar,     .     .  .  0.022 

Crown  Glass,  (green,)  .     0.036 

From  this  table  it  appears,  that  the  transparent  substances 
which  have  the  highest  dispersive  power,  are  the  oil  of  cassia 
and  the  sulphuret  of  carbon,  both  of  which  fluids  have  been 
made  to  perform  an  important  service  in  the  construction  of 
achromatic  telescopes  ;  that  flint  glass,  as  that  used  for  decan- 
ters, has  a  much  higher  dispersive  power  than  crown  glass,  pr 
that  which  is  analogous  to  window  glass;  that  the  diamond  has 
a  low  dispersive  power,  but  is  exceeded  in  this  respect  by  rock 
crystal,  the  sapphire,  and  fluor  spar,  which  last- bodies  have  the 
least  dispersive  power  of  any  known  substances. 

867.  With  these  facts  in  view,  we  may  now  inquire  by  what 
means  the  object-glass  of  the  telescope  is  rendered  achromatic. 

If  we  place  behind  LL  (Fig.  302)  a  concave  lens  GG  of  the 
same  glass,  and  having  its  surfaces  ground  to  the  same  curva- 
ture, such  a  lens  having  properties  directly  opposite  to  those  of 
th$  convex  lens,  will  neutralize  its  effects.  Consequently,  the 
rays  which  were  separated  into  their  prismatic  colors  by  the 
convex  lens,  will  be  reunited  by  the  concave  lens,  and  reproduce 
white  light.  But  though  such  a  combination  of  the  two  lenses 
will  correct  the  color,  yet  it  also  destroys  the  power  of  the  con- 
vex lens  to  form  an  image,  on  which  its  use  solely  depends. 
Could  we  find  a  concave  lens  which  would  correct  all  the  color 
and  yet  not  destroy  this  refracting  power,  the  two  lenses  would 
evidently  form  the  achromatic  combination  sought  for.  Now 
this  is  what  is  actually  done  :  by  making  the  concave  lens  of  a 
substance  which  has  a  higher  dispersive  power  than  that  of  which 
the  convex  lens  is  made,  the  curvature  of  the  concave  lens  will 
not  need  to  be  so  great  as  that  of  the  convex  lens,  and  of  course 
the  two  together,  constituting  the  compound  lens,  will  be  equiv- 
alent in  refracting  power  to  a  single  lens,  whose  convexity  is 
equal  to  the  difference  of  their  curvatures.  The  most  common 


OPTICS.  585 

combination  is  that  of  flint  glass  with  crown  glass,  the  concave 
lens  being  made  of  flint  glass,  and  the  convex  of  crown.  By  the 
table  in  Art.  866,  it  will  be  seen  that  the  dispersive  power  of 
flint  glass  is  52.  while  that  of  crown  glass  is  36,  which  numbers 
are  nearly  as  3  to  2,  and  these  numbers,  therefore,  may  be  em- 
ployed for  the  sake  of  illustration.  Since  the  power  of  the  con- 
cave lens  to  reunite  the  prismatic  rays,  is  so  much  greater  than 
that  of  the  convex  lens  to  separate  them,  we  shall  not  require  a 
refractive  power  to  effect  this  equivalent  to  that  of  the  convex 
lens  ;  that  is,  a  concave  lens  of  less  curvature  and  proportionally 
greater  focal  distance,  will  serve  our  purpose.  Therefore, 

An  achromatic  lens  is  formed  by  the  union  of  a  convex  and  a  con- 
cave lens,  whose  dispersive  powers,  are  respectively  proportional  to 
their  focal  distances. 

868.  A  telescope  furnished  with  an  object-glass  thus  formed,  is 
called  an  Achromatic  Telescope.  The  spherical  aberration  being 
corrected  by  the  methods  pointed  out  in  Art.  762,  and  the  chro- 
matic aberration  being  destroyed  in  the  manner  above  described, 
the  Refracting  Telescope  becomes  an  instrument  of  great  perfec- 
tion, and  is  reckoned  among  the  greatest  works  of  art.  Until 
recently,  it  was  rare  to  meet  with  refracting  telescopes  of  an 
aperture  of  more  than  from  three  to  five  inches.  For  we  have 
already  seen  that  the  errors  of  spherical  and  chromatic  aberra- 
tion increase  rapidly  as  the  size  of  the  aperture  is  augmented. 

If  it  be  asked,  what  is  the  use  of  a  large  aperture,  since  the 
magnifying  power  does  not  depend  upon  the  diameter  of  the  ob- 
ject-glass, but  upon  the  ratio  between  the  focal  distance  of  the 
object-glass  and  the  focal  distance  of  the  eye-glass,  (Art.  858,) 
we  answer,  that  the  use  of  a  large  aperture  is  to  admit,  con- 
dense, and  finally  convey  to  the  eye,  a  larger  beam  of  light,  and 
thus  to  render  many  objects,  as  the  smaller  stars,  or  Jupiter's 
belts,  visible,  which  otherwise  would  not  be  so,  on  account  of 
the  feebleness  of  the  light  which  they  transmit  to  us.  Want  of 
light  is  in  fact  one  of  the  greatest  difficulties  that  the  telescope 
has  to  contend  with  ;  for,  in  the  first  place,  the  object-glasses  of 
most  telescopes  are  comparatively  small,  and  are  necessarily  so 
on  account  of  the  difficulty  of  procuring  suitable  glass  for  those 
of  a  larger  size  ;  and  in  the  second  place,  of  the  light  admitted 
through  the  object-glass,  a  great  proportion  is  intercepted  and 
wasted  in  various  ways,  many  instruments  being  able  to  save 
only  the  central  rays  without  rendering  the  image  indistinct  and 
colored.  Thus,  when  very  high  magnifiers  are  applied,  (which 
of  course  have  very  small  focal  distances,)  the  rays  proceed  from 
the  focus  and  fall  upon  the  microscope  so  obliquely,  that  only 
those  which  pass  through  the  central  parts  of  the  lens  can  be 

vpd  since  such  as  fall  upon  the  marginal  parts  of  the  lens  aro 
74 


586  NATURAL   PHILOSOPHY. 

too  much  affected  by  spherical  and  chromatic  aberration,  to  form 
with  the  others  a  distinct  and  colorless  image. 

869.  Want  of  field  of  view  is  another  difficulty  to  be  surmount- 
ed.    When  we  use  an  object-glass  of  short  focus  with  a  high 
magnifier,  the  microscope  must  have  a  focus  proportionally  short, 
and  of  course  the  field  of  view  will  be  very  limited  and  the  light 
but  feeble.     This  difficulty  may  be  obviated  by  using  an  object- 
glass  of  very  great  focal  distance.     If,  for  example,  the  focal  dis- 
tance of  the  object-glass  were  only  12  inches,  in  order  to  attain 
a  magnifying  power  of  120,  we  must  employ  a  microscope  whose 
focal  distance  is  only  TV  of  an  inch.     But  if  the  focal  distance 
of  the  object-glass  were  10  feet,  or  120  inches,  then  our  micro- 
scope might  have  a  focal  distance  of  1  inch,  which  would  give  a 
larger  field  and  a  stronger  light.     With  a  view  of  obviating  sev- 
eral of  the  foregoing  difficulties,  the  earlier  astronomers  who 
used  the  telescope,  employed  for  their  object-glasses  lenses  whose 
focal  lengths  were  very  great.     Cassini,  an  Italian  astronomer, 
constructed  telescopes  eighty,  one  hundred,  and  one  hundred  and 
thirty-six  feet  long  ;  and  Huygens  employed  such  as  were  nearly 
the  same  length.    The  latter  astronomer  dispensed  with  the  tube, 
fixing  his  object-glass,  contained  in  a  short  tube,  to  the  top  of  a 
high  pole,  and  forming  the  image  in  the  air,  near  the  level  of  the 
eye,  which  image  he  viewed  with  an  eye-glass,  as  usual.     With 
telescopes  of  this  description  several  of  the  satellites  of  Saturn 
were  discovered. 

870.  But  one  of  the  most  formidable  difficulties  hitherto  en- 
countered in  the  construction  of  large  refracting  telescopes,  has 
arisen  from  the  imperfections  of  glass. 

The  difficulty  of  obtaining  glass  of  a  perfectly  homogene- 
ous composition  and  structure,  is  thus  set  forth  by  Mr.  Fara- 
day— 

;  "  Although  every  part  of  the  glass  may  in  itself  be  as  good  as 
possible,  yet  without  this  condition  [a  perfectly  homogeneous  struc- 
ture] the  parts  do  not  act  in  uniformity  with  each  other ;  the  rays 
of  light  are  deflected  from  the  course  which  they  ought  to  pur- 
sue, and  the  piece  of  glass  becomes  useless.  The  streaks,  striae, 
veins  or  tails,  which  are  seen  within  glass  otherwise  perfectly 
good,  result  from  a  want  of  this  equality  ;  they  are  visible  only 
because  they  bend  the  rays  of  light  which  pass  through  them 
from  their  rectilinear  course,  and  are  constituted  of  a  glass  hav- 
ing either  a  greater  or  a  smaller  refractive  power  than  the  neigh- 
boring parts.  When  these  irregularities  are  so  powerful  as  to 
render  their  effects  observable  by  the  naked  eye,  it  may  easily 
be  supposed  to  what  an  injurious  extent  their  influence  must  ex- 
tend, in  the  construction  of  telescopes  and'  other  instruments  of 
a  similar  nature,  where  these  faults  are  not  only  magnified  many 


OPTICS.  587 

times,  but  where  the  effect  is  to  give  an  equally  magnified  erro- 
neous representation  of  the  object  looked  at,  when  the  very  point 
to  be  attained  is  to  examine  that  object  with  the  utmost  accura- 
cy :  and  it  is  accordingly  found  that  these  striae  are  the  most  fa- 
tal faults  of  glass  intended  for  optical  purposes.  Besides  this, 
not  only  do  the  striae  themselves  occasion  harm,  but  there  is  ev- 
ery reason  to  believe  that  they  rarely  occur  in  glass  otherwise 
homogeneous.  Sometimes,  it  is  true,  a  grain  of  sand,  in  passing 
through,  and  at  the  same  time  dissolving  in  glass,  will  give  a 
streak  of  different  composition  from  the  rest  of  the  substance ; 
and  again,  a  bubble  ascending  may  lift  a  line  of  heavy  or  more 
refractive  matter  into  a  lighter  and  less  refractive  portion  above. 
Many  a  disk,  which  upon  the  most  careful  examination  has  ap- 
peared perfectly  free  from  striae,  and  quite  uniform,  has,  when 
worked  into  an  object-glass,  been  found  incapable  of  giving  a 
good  image,  on  account  of  the  existence  of  irregularities  in  the 
mass,  which,  though  not  sudden  or  strong  enough  to  occasion 
striae,  still  produce  a  confused  effect ;  and  if  this  happens  with 
glass  approaching  so  near  to  perfection,  it  happens  still  more  fre- 
quently, and  to  a  much  stronger  degree,  with  such  as  contains 
visible  irregularities."* 

871.  These  irregularities  are  much  more  frequent  in  flint  glass 
than  in  crown  ;  and  by  far  the  greatest  obstacle  to  be  overcome 
in  constructing  a  large  refracting  telescope,  is  to  procure  a  suit- 
able piece  of  flint  glass  for  the  concave  part  of  the  achromatic 
object-glass.  (Art.  867.)  This  want  of  uniformity  arises  chiefly 
from  the  different  specific  gravities  of  the  materials  that  compose 
the  glass.  Oxide  of  lead,  a  very  heavy  substance,  enters  into 
the  composition  of  flint  glass  to  the  amount  of  about  one-third 
of  its  weight.  The  oxide  of  lead  is  so  heavy  a  material,  and 
at  the  same  time  so  fusible,  that  it  melts  and  sinks  to  the  bot- 
tom, leaving  the  lighter  materials  to  accumulate  at  the  top ;  and 
so  imperfect  are  the  means  of  mixture,  under  ordinary  circum- 
stances, that  glass  of  very  different  specific  gravity,  is  produced 
from  the  bottom  and  the  top  of  the  same  crucible. 

These  circumstances  we  have  thought  worthy  of  being  recited, 
in  order  to  impress  on  the  mind  of  the  learner  the  formidable 
nature,  as  well  as  the  great  number,  of  the  difficulties  to  be 
overcome  in  the  construction  of  a  large  achromatic  telescope. 
Yet  they  have,  in  several  instances,  been  completely  surmounted. 
Fraunhofer  executed  two  telescopes  with  achromatic  object- 
glasses,  the  one  nine  inches  and  nine-tenths,  and  the  other  twelve 
inches  in  diameter;  and  at  the  period  of  his  death  he  was  pur- 
posing to  undertake  one  eighteen  inches  in  diameter.  That  of 
9.9  inches  aperture  was  made  for  the  Russian  government,  for 
the  use  of  the  observatory  at  Dorpat,  where,  under  the  direction 

*  Faraday,  Phil.  Trans.,  1830. 


000  NATURAL   PHILOSOPHY. 

of  M.  Struve,  a  distinguished  astronomer,  it  has  already  achieved 
numerous  valuable  discoveries  in  astronomy.  The  object-glass 
has  a  focal  length  of  14  feet. 

THE   TERRESTRIAL   OR   DAY   TELESCOPE. 

872.  As  the  Astronomical  Telescope  represents  objects  invert- 
ed, it  requires  to  be  so  modified  for  terrestrial  views,  that  objects 
may  appear  erect.  This  is  effected  by  the  addition  of  two  more 
lenses  of  similar  figure  to  that  of  the  eye-glass,  and  of  the  same 
focal  length.  The  first  of  these  additional  glasses,  forms  a  sec- 
ond image  of  the  object,  inverted  with  respect  to  the  first  image, 
and  therefore  erect  with  respect  to  the  object.  This  image  is 
viewed  by  the  second  glass  as  by  any  simple  microscope.  Thus, 
AB,  the  object-glass,  forms  an  inverted  image  nm  of  the  object 
MN.  Instead  of  viewing  this  image  by  the  eye  placed  at  I>,  as 
in  the  common  astronomical  telescope,  we  suffer  the  pencils  of 
parallel  rays  to  cross  each  other  at  L  and  fall  upon  a  second  lens 


EF  (similar  in  all  respects  to  CD)  which  collects  them  into  an 
image  m'n'  in  its  focus  of  parallel  rays,  which  image  is  viewed 
by  the  eye-glass  GH  in  the  same  manner  as  the  object  itself 
would  be. 

As  some  portion  of  the  light  is  reflected,  and  some  absorbed 
and  dissipated  by  passing  through  these  additional  lenses,  they  of 
course  diminish  the  brightness  of  the  view ;  but  in  the  daytime 
there  will  usually  be  light  enough  for  distinct  vision  after  this 
loss  is  sustained,  while  it  is  more  agreeable  and  convenient  to 
have  the  objects  presented  to  us  in  their  natural  positions  than 
inverted.  It  will  be  remarked  that  the  additional  lenses  do  not 
magnify,  the  focal  length  of  each  being  the  same  as  that  of  the 
first  eye-glass.  Were  they  rendered  smaller  for  the  purpose  of 
magnifying,  the  field  of  view  and  the  light  would  both  be  im- 
paired. 

873.  We  usually  find  in  telescopes,  particularly  those  designed 
for  terrestrial  objects,  some  contrivance,  as  a  draw-tube,  by  which 
the  eye-glass  can  be  brought  nearer  to.  or  withdrawn  from  the 
object-glass.  This  is  to  accommodate  the  instrument  to  objects, 
at  different  distances.  When  it  is  directed  to  very  near  objects, 
the  image  is  thrown  further  back,  and  therefore  in  order  that  it 
may  be  in  the  focus  of  the  eye-glass,  (which  is  essential  to  dis- 
tinct vision,)  the  latter  must  be  drawn  backward ;  but  where  the 


OPTICS.  589 

V 

object  is  remote,  the  image  is  formed  nearer  to  the  object-glass, 
and  then  the  eye-glass  must  be  moved  forward,  till  its  focus  of 
parallel  rays  comes  to  the  place  of  the  image.  For  a  similar 
reason,  near-sighted  persons  require  the  eye-glass  to  be  brought 
nearer  than  usual  to  the  object-glass ;  for  then  the  image  will  be 
nearer  to  the  eye-glass  than  its  focus  of  parallel  rays,  and  the 
rays  will  meet  the  eye  diverging,  a  condition  favorable  to  eyes 
naturally  too  convex.  For  a  contrary  reason,  long-sighted  per- 
sons, who  usually  wear  convex  spectacles,  may  adjust  the  tele- 
scope to  suit  their  eyes  without  spectacles,  by  removing  the  eye- 
glass further  back  than  usual. 

Most  terrestrial  telescopes  contain  a  greater  number  of  glasses 
than  are  represented  in  figure  303.  Such  a  number  are  used  for 
the  purpose  of  correcting  spherical  and  chromatic  aberration, 
these  errors  being  less  in  several  flat  and  thin  lenses  than  in  a 
smaller  number  of  equivalent  lenses  of  greater  curvature. 

Astronomical  telescopes  are  easily  adapted  to  terrestrial  ob- 
servations, by  removing  the  eye-glass  and  substituting  a  tube 
containing  the  additional  glasses  for  rendering  the  view  erect. 


GALILEO  S    TELESCOPE. 

874.  This  instrument  was  the  first  astronomical  telescope,  hav- 
ing been  invented  by  Galileo,  as  the  name  implies.  It  differs 
from  the  common  astronomical  telescope,  in  having  for  the  eye- 
glass a  concave  instead  of  a  convex  lens,  which  receives  the  pen- 
cils of  light,  as  they  are  converging  to  form  an  image,  at  such  a 
distance  from  the  focus  to  which  they  tend,  as  to  render  them 
parallel.  Thus,  the  object-glass  AB  collects  the  rays  of  light  as 
they  proceed  from  the  object  MN,  and  makes  them  converge  to- 
ward the  focus  at  E.  But  the  concave  lens  CD  is  interposed  at 
such  a  point  as  to  render  these  converging  rays  parallel,  and  in 
this  way  they  come  to  the  eye  situated  behind  the  lens. 


Since  the  concave  lens  restores  the  rays  to  that  state  of  paral- 
lelism which  they  had  before  they  passed  through  the  object- 
glass,  the  learner  may  not  readily  see  how  this  instrument  aids 
the  eye.  That  it  does  so,  however,  will  be  apparent  from  the 
following  considerations. 

First,  a  much  broader  beam  of  light  falls  upon  the  object-glass 
than  upon  the  naked  pupil  of  the  eye,  the  greater  part  of  which 


590  NATURAL   PHILOSOPHY. 

is  collected  and  conveyed  to  the  eye.  By  this  means  the  bright- 
ness of  objects  is  greatly  increased. 

•Secondly,  as  in  the  astronomical  telescope,  (Art.  857,)  were  the 
eye  situated  at  the  center  of  the  object-glass,  the  object  and  the 
image  formed  by  the  object-glass  would  have  the  same  apparent 
dimensions  ;  and  inasmuch  as  the  eye-glass  enables  us  to  view 
this  image  much  nearer,  it  increases  its  apparent  dimensions  in 
the  same  ratio.  But  when  we  use  a  concave  lens  situated  as  in 
the  Galilean  telescope,  the  effect  is  the  same  as  that  of  a  convex 
lens  situated  in  the  same  manner  on  the  other  side  of  the  focus, 
so  that  the  rays  would  come  to  the  eye  parallel.  Hence,  in  the 
Galilean  as  in  the  common  astronomical  telescope,  the  magnify- 
ing power  is  as  the  ratio  of  the  focal  distance  of  the  object-glass 
to  that  of  the  eye-glass. 

This  form  of  the  telescope  has  several  advantages  and  several 
disadvantages,  when  compared  with  the  ordinary  form.  In  the 
first  place,  requiring  but  two  glasses  to  present  objects  erect,  it 
occasions  less  loss  of  light  than  the  ordinary  form,  and  presents 
objects  with  proportionally  greater  brightness.  In  the  second 
place,  the  eye-glass  being  between  the  object-glass  and  the  image, 
instead  of  beyond  it,  the  instrument  admits  of  being  made  short 
and  compact,  a  circumstance  which  fits  it  for  the  purposes  of  an 
opera-glass,  to  which  use  it  is  frequently  applied.  In  the  third 
place,  the  concave  lens  corrects  the  chromatic  aberration  of  the 
convex  lens,  and  where  a  proper  proportion  is  observed  between 
the  curvatures  of  the  two  lenses,  the  instrument  is  easily  ren- 
dered achromatic.  The  chief  disadvantage  attending  the  instru- 
ment, is  its  limited  field  of  view.  For  the  pencils  of  parallel  rays, 
after  passing  through  the  concave  eye-glass,  diverge  from  one 
another,  those  toward  the  marginal  parts  of  the  lens  being  turned 
from  those  that  are  contiguous  to  the  axis,  and  therefore  not  en- 
tering the  pupil  of  the  eye.  And  since  only  those  near  the  axis 
at  E,  (Fig.  304,)  can  enter  the  pupil,  the  field  of  view  must  de- 
pend on  the  dimensions  of  the  pupil,  and  cannot  be  increased  by 
increasing  the  length  of  the  instrument,  as  in  the  refracting  tel- 
escope. This  defect  has  caused  this  kind  of  telescope  to  fall  in- 
to disuse  for  astronomical  purposes. 


REFLECTING   TELESCOPE. 

875.  Reflecting  Telescopes  differ  in  principle  from  those  al- 
ready described  only  in  forming  their  image  by  a  concave  reflector 
instead  of  a  convex  object-glass.  The  most  common  form  of  the 
Reflecting  Telescope  is  the  Gregorian,  so  called  from  the  in- 
ventor, Dr.  James  Gregory,  of  Scotland.  The  general  principles 
of  this  instrument  may  be  explained  as  follows  : 

In  the  Gregorian  Telescope,  the  light  (supposed  to  come  in 


OPTICS. 


591 


parallel  rays)  is  first  received  by  a  large  concave  speculum,  by 
which  it  is  brought  to  a  focus,  and  made  to  form  an  inverted  im- 
age. On  the  opposite  side  of  this  image,  and  facing  the  large 
speculum,  is  placed  a  small  concave  speculum,  of  greater  curva- 
ture, at  such  a  distance  from  the  image  that  the  rays  proceeding 
from  it  and  falling  on  the  speculum  are  made  to  converge  to  a 
focus  situated  a  small  distance  behind  the  large  speculum,  pass- 
ing through  a  circular  aperture  in  the  center  of  it.  This  second 
image  is  magnified  by  a  microscope,  as  in  the  Refracting  Tele- 
scope. This  description  may  now  be  applied  to  the  annexed 
figure. 

ABCD,  a  large  open  tube  of  brass,  iron,  or  mahogany,  to  con- 
tain the  reflectors. 

abed,  a  smaller  tube,  to  receive  the  second  image  and  the  eye- 
glass. 

EE,  a  large  concave  speculum,  usually  composed  of  a  metallic 
compound  called  speculum  metal. 

Fig.  305. 


FF,  small  concave  speculum. 

mn,  image  formed  by  the  large  reflector. 

nm,  image  formed  by  the  small  reflector. 

G,  eye-glass. 

WY,  a  metallic  rod  having  a  screw  connected  with  the  small 
reflector,  by  means  of  which  this  reflector  is  made  to  approach 
the  first  image,  or  to  recede  from  it. 

Some  of  the  pencils  of  rays  necessary  to  form  the  respective 
images  are  omitted  in  the  figure,  to  prevent  confusion. 

876.  From  the  foregoing  construction  it  is  evident,  first,  that 
the  image  viewed  by  the  eye  being  in  the  same  position  with 
the  object,  the  latter  will  appear  erect ;  secondly,  that  since  the 
mirrors  may  be  formed  of  a  parabolic  figure,*  all  spherical  aber- 
ration may  be  easily  prevented,  (Art.  761  ;)  thirdly,  that  since 
light  is  not  decomposed  by  reflexion,  reflecting  telescopes  are  not 
subject  to  chromatic  aberration ;  and,  hence,  that  it  is  not  neces- 

»  An  elliptical  figure  has  the  same  property. 


592  NATURAL,    PHILOSOPHY. 

sary  to  lengthen  the  tube  as  the  aperture  is  increased,  as  is 
the  case  in  refracting  telescopes ;  but  since  the  light  \vill  de- 
pend, chiefly,  on  the  size  of  the  large  reflector,  a  strong 
light  may  be  obtained  with  a  comparatively  short  tube.  The 
achromatic  telescope,  however,  with  all  the  latest  improvements, 
is  deemed  a  more  perfect  and  more  convenient  instrument  than 
the  reflecting  telescope ;  and  it  is  supposed  that  there  will  be  no 
occasion  hereafter  to  construct  reflectors  of  such  enormous  dimen- 
sions as  those  of  Dr.  Herschel.  Some  account  of  his  forty  feet 
reflector  may  form  a  suitable  close  to  this  sketch  of  optical  in- 
struments. 

877.  Under  the  munificent  patronage  of  George  III,  Sir  Wil- 
liam Herschel  began,  in  1785,  to  construct  a  telescope  forty  feet 
long,  and  in  1789,  on  the  day  when  it  was  completed,  he  discov- 
ered with  it  the  sixth  satellite  of  Saturn.  The  great  speculum 
was  more  than  four  feet  in  diameter,  and  weighed  two  thousand 
one  hundred  and  eighteen  pounds.  Its  focal  length  was  forty 
feet.  The  tube  which  contained  it  was  made  of  sheet  iron. 

The  light  afforded  by  this  instrument  was  astonishingly  great. 
The  largest  fixed  stars,  as  Sirius,  shone  in  it  with  the  splendor 
of  the  sun.  The  reason  of  this  will  be  obvious,  when  we  reflect 
that  it  collected  and  conveyed  to  the  eye,  in  the  place  of  the 
small  beam  that  enters  the  naked  organ,  a  beam  of  light  from 
the  sl^ar  more  than  four  feet  in  diameter.  Hence,  it  was  suited 
to  reveal  to  the  eye  numberless  stars  and  clusters  of  stars,  which 
preceding  telescopes  had  failed  to  exhibit,  because  they  could 
not  collect  a  sufficient  quantity  of  their  light.  To  economize  the 
light  to  the  best  advantage,  the  small  mirror  employed  in  the 
Gregorian  telescope,  (see  Fig.  305,)  was  dispensed  with,  since 
every  successive  reflexion  dissipates  a  considerable  portion  of  the 
light,  and  the  image  was  thrown  near  to  the  open  mouth  of  the 
tube,  where  it  was  viewed  by  the  eye-glass  directly,  the  observer 
being  seated  so  as  to  look  into  the  mouth  in  front.  In  order  to 
prevent  the  head  from  obstructing  too  much  of  the  light,  the 
image  was  formed  near  one  side  of  the  tube.*  Its  greatest  mag- 
nifying power  was  six  thousand  four  hundred  and  fifty ;  but  this 
was  used  only  for  the  smallest  stars. 

The  greatest  telescope  hitherto  constructed,  is  the  one  recent- 
ly built  in  Ireland  by  Lord  Oxmanton,  (Earl  of  Ross,)  which  has 
a  focal  distance  of  50  feet,  and  an  aperture  of  6  feet.  This 
gigantic  instrument,  it  is  expected,  will  reveal  to  us  many  secrets 
of  the  skies  hitherto  hidden  from  human  view. 

*  Holcomb's  telescopes  are  constructed  in  the  same  manner. 
FINIS. 


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RECOMMENDATIONS  OP  "COFFIN'S  CONIC  SECTIONS.' 


FROM  PROF.  MASON,  BETHANY  COLLEGE,  VA. 

I  have  examined  "  Coffin's  Conic  Sections  and  Analytical  Geometry,"  and  hesitate  not 
to  say,  that  its  size,  plan  and  general  arrangement,  ought  to  procure  for  it  a  place  in  th« 
hands  of  every  teacher  who  wishes  to  give  a  liberal  knowledge  of  Mathematics,  without 
abridging  the  other  branches  of  a  good  education.  I  am  happy  to  say  that  your  book  ia 
better  adapted  to  my  present  wants  than  any  I  have  seen.  I  have  therefore  concluded 
to  adopt  it  for  my  next  class. 

FROM  J.  S.  LEBAR,  PRINCIPAL  OF  THE  HIGH  SCHOOL,  HACKETTSTOWN,  N.  J. 
By  bringing  out  this  work  you  have  advanced  the  cause  of  science,  merited  tho  thanks 

>f  the  lovers  of  learning,  and  done  a  lasting  good  to  the  public As  a  general  principle, 

f  am  opposed  to  the  shortening  of  books  in  order  to  shorten  the  course  of  study.  The  spirit 
)f  the  day  is  to  get  along  most  rapidly,  not  most  thoroughly ;  by  bestowing  the  least  pos- 
sible labor,  both  in  physical  and  mathematical  science.  But  from  the  examination  I  have 
been  able  to  make  of  your  work,  I  find  it  contains  some  important  matter,  not  found  in 
other  works  of  a  like  character,  whilst  nothing  that  is  valuable  is  omitted.  The  compass  is 
contracted,  but  the  whole  circle  is  there.  It  is  just  the  work  we  need,  and  I  shall  adopt  it 

in  our  Academy  in  preference  to  other  treatises  upon  these  subjects Certainly  by 

your  combining  the  geometrical  and  analytical  methods,  you  present  the  subject  more 

clearly Your  demonstrations  are  admirable  ;  they  are  so  concise,  yet  so  simple,  that 

no  student  can  pass  over  them  without  fully  comprehending  them. 

FROM  PROF.  SCDLER,  DICKINSON  COLLEGE,  PENN. 

I  have  received  and  read  with  great  pleasure  your  "  Elements  of  Conic  Sections  and 
Analytical  Geometry,"  and  cannot  hesitate  to  commend  it  as  an  excellent  system.  It 
embraces  in  a  parallel  view,  the  Geometrical  Theory  and  the  Analytical  investigation  of 
the  properties  of  the  Conic  Sections — the  method  which  I  have  always  pursued  in  the 
instruction  in  my  department,  but  have  felt  the  want  of  a  proper  text-book  on  the  subject. 
This  want  has  been  met  by  your  Treatise,  and  I  have  adopted  it  as  a  text-book  with 
pleasure. 

FROM  PROF.  FOUCHE,  ST.  JOHN'S  COLLEGE,  N.  Y. 

Your  work  was  examined,  I  do  not  say  by  me,  but  by  judges  more  competent  than  I 
am.  We  are  unanimous  in  saying  that  your  demonstrations,  with  respect  to  perspicuity, 
aie  to  be  highly  praised.  You  are  clear,  concise,  and,  whilst  you  avoid  obscurity,  you 
find  the  means  of  being  both  exact  and  brief.  This  is  no  small  merit,  and  you  are  enti- 
tled to  the  gratitude  of  students,  to  whom  you  spare  useless  difficulties  and  exertions. 


FROM  PROF.  LOOMIS,  COLLEGE  OF  NEW  JERSEY,  PRINCETON,  N.  J. 
Your  treatise  on  "  Analytical  Geometry"  appears  to  me  to  possess  advantages  over 
any  other  treatise  which  I  have  examined.  One  of  these  advantages  I  consider  to  be, 
the  division  into  Propositions  distinctly  enunciated,  so 'as  to  keep  constantly  before  the 
mind  of  the  student  the  object  of  his  investigation.  Another  advantage  consists  in  the 
introduction  of  numerical  examples,  which  afford  a  useful  exercise  to  the  student,  and 
present  the  best  test  of  the  clearness  of  his  conceptions. 


FROM  J.  S.  ALVERSON,  PRINCIPAL  OF  THE  GENESEE  WESLEYAN  SEMINARY,  N.  Y. 
I  have  received  and  examined  a  copy  of  your  recent  work  on  "  Conic  Sections."  .... 
The  manner  in  which  you  have  introduced  both  the  geometrical  and  analytical  demon- 
strations, ought  to  secure  success  to  your  work.     You  present  what  I  think  is  needed  by 
the  American  student     In  the  selection  of  problems  and  in  the  elucidation  of  parts  that 

often  prove  unreasonably  obscure  to  beginners,  you  have  been  very  happy I  recom- 

mend  the  work  with  confidence  to  the  attention  of  Instructors  and  Students. 


RECOMMENDATIONS. 

FROM  PROF.  STEVENS,  OAKLAND  COLLEGE,  Miss. 

I  sent  to  New  Orleans  for  a  copy  of  your  treatise  on  the  "  Conic  Sections  and  Analyt- 
ical Geometry,"  which  I  have  received  and  have  examined  with  some  care  and  much 
satisfaction.  I  think  after  the  perusal  I  have  given  it,  I  may  safely  pronounce  your  trea- 
tise eminently  suited  for  use  in  colleges ;  it  is  sufficiently  elaborate,  the  arrangement  is 
good,  and  there  is  a  plainness  and  simplicity  about  it  which  I  admire  very  much. 


MESSRS.  COLLINS  &  BROTHER. — Gent:  Illness  has  prevented  me  from  examining 
Prof.  Coffin's  "  Treatise  on  Conic  Sections"  till  very  recently.  I  am  greatly  pleased  with 
the  work,  and  think  that  the  author  has  succeeded  in  developing  the  most  interesting  and 
important  properties  of  these  curves  with  unusual  simplicity  and  brevity  in  the  demon- 
strations. To  this,  the  common  property  of  the  curves,  assumed  as  their  fundamental 
characteristic,  is,  by  the  skill^  and  tact  of  the  author,  made  to  contribute  very  much. 
This  property,  while  it  unites  the  three  curves  in  a  common  bond,  and  gives  them  a 
common  source,  is  that  on  which  the  more  use'ul  properties  seem  most  immediately  to 
depend,  and  from  which  they  are  most  readily  and  naturally  deduced.  In  the  Geometri- 
cal portion  there  is  a  very  just  medium  preserved,  in  the  extent  to  which  the  discussion 
of  the  curves  is  carried,  the  investigations  being  limited  to  the  most  essential  and  practi- 
cally useful  properties,  and  requiring  no  useless  expenditure  of  time  on  points  merely  spec- 
ulative or  curious.  The  second  part,  devoted  to  "  Analytical  Geometry,"  is  a  very  im- 
portant addition  to  the  work,  and  one,  I  think,  without  which  the  student  will  be  very 
illy  prepared^  ttrmake  his  knowledge  of  these  curves,  readily  and  extensively  available  in 
their  application  to  astronomical  investigations.  The  work  I  regard  as  a  very  decided 
improvementa)n  'the  old  systems  and  treatises,  and  think  in  its  preparation  and  publication 
a  very  valuagle'jboutribution  has  been  made  to  educational  instrumentalities. 

Very  respectfully  yours, 

WESLEY  AN  UNIVERSITY.  April  20,  1849.  AUG.  W.  SMITH. 

FROM  THE  METHODIST  QUARTERLY  REVIEW. 
"  Elements  of  the  Conic  Sections  and  Analytical  Geometry,  by  JAMES  H.  COFFIN,  A.  M. 

Professor  of  Mathematics,  tyc.,  in  Lafayette  College." 

In  this  treatise  the  doctrine  of  the  Conic  Sections  is  taught  first  geometrically  ;  and  in 
a  second  part,  the  student  is  taught  how  to  represent  lines,  curves,  and  surfaces  analyti- 
cally, and  to  solve  problems  relating  to  them.  We  have  examined  the  work  with  care, 
and  testify  to  the  skill,  tact,  and  neatness  of  its  expositions.  Most  books  of  Analytical 
Geometry  are  blind  to  scholars ;  most  of  them  never  learn,  unless  they  have  a  teacher 
unusually  skilful  and  diligent,  how  to  interpret  algebraical  expressions,  or  how  to  make 
practical  use  of  equations.  It  is  precisely  for  its  clearness,  its  practical  character,  and  its 
adaptation  to  the  work  of  the  recitation  room,  that  wo  heartily  commend  this  volume. 


FROM  J.  F.  JENKINS,  PRINCIPAL  OF  THE  NORTH  SALEM  ACADEMY. 

I  have  recently  had  an  opportunity  of  examining  your  treatise  on  "  Conic  Sections  and 
Analytical  Geometry,"  and  am  happy  to  state  that,  in  my  opinion,  it  i*  a  work  of  much 
merit.  The  selection  and  arrangement  of  the  Propositions  appear  to  me  judicious ;  and 
the  definition  which  you  adopt,  at  the  commencement,  has  certainly  the  advantage  of 
supplying  more  simple  and  obvious  demonstrations,  in  place  of  those  whose  prolixity  and 
abstruseness  have  hitherto  rendered  this  very  important  portion  of  Mathematics  so  for- 
midable to  learners.  The  first  part  appears  to  contain  all  the  properties  of  the  Sections 
which  are  required  in  the  branches  of  Mathematical  Philosophy  usually  embraced  in  our 
College  courses ;  and  the  second  part  furnishes  a  valuable  introduction  to  Analytical  Ge- 
ometry, rendered  more  valuable  by  its  immediate  connection  with  what  precedes,  and  by 
the  practical  nature  of  the  problems  proposed  for  investigation  by  analysis.  I  intend  in- 
troducing it  as  a  text-book  in  this  Institution. 


FROM  JAMES  T.  DORAN,  TEACHER,  MANALAPAN,  N  J 
I  think  it  decidedly  the  best  work  in  print  upon  the  sublet. 


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